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[Axiom-developer] 20081206.01.tpd.patch (regression file fixed created)
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20081206.01.tpd.patch (regression file fixed created) |
|
Date: |
Sun, 7 Dec 2008 07:04:24 -0600 |
This adds 267 new tests to the Axiom regression test suite.
======================================================================
diff --git a/changelog b/changelog
index beaca4a..02caa58 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,12 @@
+20081206 tpd src/axiom-website/patches.html 20081206.01.tpd.patch
+20081206 tpd src/input/Makefile add fixed.regress
+20081206 tpd src/input/fixed.input convert to a regression file
+20081206 tpd src/input/schaum17.input fix missing space typo
+20081205 tpd src/axiom-website/patches.html 20081205.02.tpd.patch
+20081205 txl src/input/schaum17.input fix 14.355, 14.356
+20081205 tpd src/interp/setq.lisp add Tim Lahey
+20081205 tpd readme add Tim Lahey
+20081205 txl Tim Lahey <address@hidden>
20081205 tpd src/axiom-website/patches.html 20081205.01.tpd.patch
20081205 tpd src/algebra/integer.spad fix r21bugsbig regression
20081205 tpd src/input/r20bugs.input.pamphlet fix regression
diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html
index 1891268..2428e91 100644
--- a/src/axiom-website/patches.html
+++ b/src/axiom-website/patches.html
@@ -785,6 +785,8 @@ clean up input regression files<br/>
bug 7161: integer.spad remove signature change<br/>
<a href="patches/20081205.02.tpd.patch">20081205.02.tpd.patch</a>
schaum17 fix 14.355, 14.356<br/>
+<a href="patches/20081206.01.tpd.patch">20081206.01.tpd.patch</a>
+regression file fixed created<br/>
</body>
</html>
\ No newline at end of file
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet
index 032ce85..2cf453c 100644
--- a/src/input/Makefile.pamphlet
+++ b/src/input/Makefile.pamphlet
@@ -305,7 +305,8 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress
alist.regress \
expexpan.regress explim.regress expr1.regress exprode.regress \
expr.regress exseries.regress exsum.regress exprpoly.regress \
farray.regress ffdemo.regress fferr.regress ffx72.regress \
- fib.regress file.regress float1.regress float2.regress \
+ fib.regress file.regress fixed.regress \
+ float1.regress float2.regress \
float.regress fname1.regress fname.regress fnla.regress \
fns.regress fparfrac.regress fparfrc.regress fr1.regress \
fr2.regress frac.regress fr.regress free.regress \
@@ -315,7 +316,7 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress
alist.regress \
herm.regress heugcd.regress \
hexadec.regress ico.regress ideal.regress \
ifact.regress ifthenelse.regress \
- infprod.regress intaf.regress intbypart.regress \
+ infprod.regress intaf.regress intbypart.regress \
intdeq.regress \
intef2.regress intef.regress intg0.regress intheory.regress \
intmix2.regress intmix.regress int.regress intrf.regress \
diff --git a/src/input/fixed.input.pamphlet b/src/input/fixed.input.pamphlet
index c221351..4a5a5c3 100644
--- a/src/input/fixed.input.pamphlet
+++ b/src/input/fixed.input.pamphlet
@@ -10,413 +10,1181 @@
\tableofcontents
\eject
\section{License}
-<<license>>=
---Copyright The Numerical Algorithms Group Limited 1996.
-@
+Copyright The Numerical Algorithms Group Limited 1996.
+
+daly/10/06/92
<<*>>=
-<<license>>
---fixed.input
+)spool fixed.output
+)set message test on
+)set message auto off
+)set break resume
+)set expose add constructor SquareMatrix
--------------------------------------------------------------------
--- daly/10/06/92
--------------------------------------------------------------------
+@
+Something is wrong with the integration of serices, when the
+coeffiennts are not constants. while the series is mathematically
+correct, this is not what one expects as the taylor series, which
+does not exist.
+<<*>>=
)clear all
--- something is wrong with the integration of serices, when the
--- coeffiennts are not constants. while the series is mathematically
--- correct, this is not what one expects as the taylor series, which
--- does not exist.
-series(x/(x+log(x)))
--- here integrate is treating log(x) as a constant, which is incorrect.
-integrate(%)
--------------------------------------------------------------------
--- bmt/10/19/92 integrate problem -- serious, wrong answer
--------------------------------------------------------------------
+--S 1
+t1:=series(x/(x+log(x)))
+--R
+--R (1)
+--R 1 1 2 1 3 1 4 1 5 1
6
+--R ------ x - ------- x + ------- x - ------- x + ------- x - -------
x
+--R log(x) 2 3 4 5 6
+--R log(x) log(x) log(x) log(x) log(x)
+--R +
+--R 1 7 1 8 1 9 1 10 1 11
12
+--R ------- x - ------- x + ------- x - -------- x + -------- x + O(x
)
+--R 7 8 9 10 11
+--R log(x) log(x) log(x) log(x) log(x)
+--R Type: GeneralUnivariatePowerSeries(Expression
Integer,x,0)
+--E
+
+@
+Here integrate is treating log(x) as a constant, which is incorrect.
+<<*>>=
+
+--S 2
+integrate(t1)
+--R
+--R
+--RDaly Bug
+--R >> Error detected within library code:
+--R "'integrate' unavailable on this domain; use 'approximate' first"
+--R
+--R Continuing to read the file...
+--R
+--E
+
+@
+bmt/10/19/92 integrate problem -- serious, wrong answer
+<<*>>=
)clear all
-2*sin(t)*sqrt(1+cos(t))
---correct answer is 2*sin(t)/sqrt(1+cos(t))
-integrate(%,t)
--------------------------------------------------------------------
--- bmt/10/20/92 multiple complex in type tower
--------------------------------------------------------------------
+--S 3
+t1:=2*sin(t)*sqrt(1+cos(t))
+--R
+--R +----------+
+--R (1) 2sin(t)\|cos(t) + 1
+--R Type: Expression
Integer
+--E
+
+@
+Correct answer is \[\frac{2\sin(t)}{\sqrt{1+\cos(t)}}\]
+<<*>>=
+
+--S 4
+integrate(t1,t)
+--R
+--R +----------+
+--R (- 4cos(t) - 4)\|cos(t) + 1
+--R (2) ----------------------------
+--R 3
+--R Type: Union(Expression
Integer,...)
+--E
+
+@
+bmt/10/20/92 multiple complex in type tower
+<<*>>=
)clear all
+
+--S 5
n:Complex ?
--- an invalid type. can't have 2 complex constructors in a tower
+--R Type:
Void
+--E
+
+@
+An invalid type. can't have 2 complex constructors in a tower
+<<*>>=
+
+--S 6
n:=x/y+%i
+--R
+--R x
+--R (2) - + %i
+--R y
+--R Type: Complex Fraction Polynomial
Integer
+--E
+
+@
+grabm/09/28/92 coercion bug
--------------------------------------------------------------------------
--- grabm/09/28/92 coercion bug
--- comment: this coercion must carefully check whether there is a symbol
--- among the kernels which is equal to an indeterminate of the
--- polynomial ring, and not simply considering every expression
--- as a constant
--------------------------------------------------------------------------
+This coercion must carefully check whether there is a symbol
+among the kernels which is equal to an indeterminate of the
+polynomial ring, and not simply considering every expression
+as a constant
+<<*>>=
)clear all
+
+--S 7
f:=(a-b-c-d)**2::EXPR INT
-f::DMP(['a,'b],EXPR INT)
-degree %
-
--------------------------------------------------------------------
--- bmt/10/26/92 wrong answer
--- comment:I believe this problem simplifies to
--- lfintegrate(sqrt(u**3+u**2),u) which returns the
--- wrong answer due to some confusion in prootintegrate in INTPAF.
--- I think the confusion happens with the use of radPoly and rootPoly.
--- The answer is computed with respect to the result returned by rootPoly
--- but the kernel which is substituted for "y" corresponds to radPoly.
--------------------------------------------------------------------
-)clear all
--- should be 2*sin(t)/sqrt(1+cos(t))
+--R
+--R 2 2 2 2
+--R (1) d + (2c + 2b - 2a)d + c + (2b - 2a)c + b - 2a b + a
+--R Type: Expression
Integer
+--E
+
+--S 8
+t1:=f::DMP(['a,'b],EXPR INT)
+--R
+--R 2 2 2 2
+--R (2) a - 2a b + (- 2d - 2c)a + b + (2d + 2c)b + d + 2c d + c
+--R Type: DistributedMultivariatePolynomial([a,b],Expression
Integer)
+--E
+
+--S 9
+degree t1
+--R
+--R (3) [2,0]
+--R Type:
DirectProduct(2,NonNegativeInteger)
+--E
+
+@
+bmt/10/26/92 wrong answer
+
+I believe this problem simplifies to
+{\tt lfintegrate(sqrt(u**3+u**2),u)} which returns the
+wrong answer due to some confusion in prootintegrate in {\tt INTPAF}.
+I think the confusion happens with the use of radPoly and rootPoly.
+The answer is computed with respect to the result returned by rootPoly
+but the kernel which is substituted for ``y'' corresponds to radPoly.
+
+The answer should be should be \[\frac{2*\sin(t)}{\sqrt{1+\cos(t)}}\]
+<<*>>=
+)clear all
+
+--S 10
integrate(sqrt(1+cos(x)),x)
+--R
+--R +----------+
+--R 2sin(x)\|cos(x) + 1
+--R (1) --------------------
+--R cos(x) + 1
+--R Type: Union(Expression
Integer,...)
+--E
--------------------------------------------------------------------
--- bmt/09/28/92 bug in ITRIGMNP
--- gives:
--- Cannot convert kernel to gaussian function
--------------------------------------------------------------------
+@
+bmt/09/28/92 bug in ITRIGMNP
+
+gives: Cannot convert kernel to gaussian function
+<<*>>=
)clear all
+
+--S 11
integrate(exp(x**2),x)
+--R
+--R x 2
+--I ++ %K
+--I (1) | %e d%K
+--R ++
+--R Type: Union(Expression
Integer,...)
+--E
--------------------------------------------------------------------
--- themos/11/06/92 integration bug
---comment:
---I suggest changing line 11 of limitedLogs in INTTR from:
--- rows := max(degree den, 1 + "max"/[degree(u.contrib) for u in l1])
---to:
--- rows := max(degree den,
--- 1 + reduce(max,[degree(u.contrib) for u in l1],0)$List(N))
---Does you agree, thus max/ empty list would return 0.
--------------------------------------------------------------------
+@
+themos/11/06/92 integration bug
+
+I suggest changing line 11 of limitedLogs in INTTR from:
+\begin{verbatim}
+ rows := max(degree den, 1 + "max"/[degree(u.contrib) for u in l1])
+\end{verbatim}
+to
+\begin{verbatim}
+ rows := max(degree den,
+ 1 + reduce(max,[degree(u.contrib) for u in l1],0)$List(N))
+\end{verbatim}
+Do you agree, thus max/ empty list would return 0.
+<<*>>=
)clear all
+
+--S 12
f:=log(1-(b*x/(a+c*x**2)))/x
--- 2
--- c x - b x + a
--- log(--------------)
--- 2
--- c x + a
--- (1) -------------------
--- x
-integrate(%,x)
--- 2
--- %J c - %J b + a
--- log(---------------)
--- x 2
--- ++ %J c + a
--- (2) | -------------------- d%J
--- ++ %J
-expand f
--- 2 2
--- - log(c x + a) + log(c x - b x + a)
--- (3) -------------------------------------
--- x
-integrate(%,x)
--- >> Error detected within library code:
--- No identity element for reduce of empty list using operation
--- max
-
--------------------------------------------------------------------
--- bmt/11/17/92 interpreter resolve problem
--------------------------------------------------------------------
--- says cannot find an appropriate *
+--R
+--R 2
+--R c x - b x + a
+--R log(--------------)
+--R 2
+--R c x + a
+--R (1) -------------------
+--R x
+--R Type: Expression
Integer
+--E
+
+--S 13
+integrate(f,x)
+--R
+--R 2
+--I %K c - %K b + a
+--R log(---------------)
+--R x 2
+--I ++ %K c + a
+--I (2) | -------------------- d%K
+--I ++ %K
+--R Type: Union(Expression
Integer,...)
+--E
+
+--S 14
+g:=expand f
+--R
+--R 2 2
+--R - log(c x + a) + log(c x - b x + a)
+--R (3) -------------------------------------
+--R x
+--R Type: Expression
Integer
+--E
+
+--S 15
+integrate(g,x)
+--R
+--R x 2 2
+--I ++ - log(%K c + a) + log(%K c - %K b + a)
+--I (4) | -------------------------------------- d%K
+--I ++ %K
+--R Type: Union(Expression
Integer,...)
+--E
+
+@
+bmt/11/17/92 interpreter resolve problem\\
+Says cannot find an appropriate *
+<<*>>=
)clear all
+
+--S 16
%i/m*sqrt(m)
+--R
+--R +-+
+--R %i\|m
+--R (1) ------
+--R m
+--R Type: Expression Complex
Integer
+--E
--------------------------------------------------------------------------
--- tpd/09/22/92 this generates extra brackets
--------------------------------------------------------------------------
+@
+tpd/09/22/92 this generates extra brackets
+<<*>>=
)clear all
+
+--S 17
foo n == matrix[[i for i in 1..n] for j in 1..n]
+--R Type:
Void
+--E
+
+--S 18
foo
+--R
+--R (2) foo n == [[i for i in 1..n] for j in 1..n]
+--R Type: FunctionCalled
foo
+--E
+
+)clear all
+
+msq := Matrix SquareMatrix(2,POLY INT)
--------------------------------------------------------------------------
---- tpd/09/24/92 matrix multiplication appears broken in the product.
--- fails with
--- >> System error:
--- The function |deleteAssoc| is undefined.
--- You are being returned to the top level of the interpreter.
--- comment: (barry)
--- This bug is due to the new autoloading scheme for the compiler, the
--- function deleteAssoc needs to be moved into the in core system.
--- As a work around,
--- if you compile a non existent file, the apropriate files will be loaded,
--- e.g.:
--- )co foo
--- comment: this works in version 2
--------------------------------------------------------------------------
+@
+tpd/09/24/92 matrix multiplication appears broken in the product.\\
+fails with
+\begin{verbatim}
+ >> System error:
+ The function |deleteAssoc| is undefined.
+ You are being returned to the top level of the interpreter.
+\end{verbatim}
+comment: (barry)\\
+This bug is due to the new autoloading scheme for the compiler, the
+function deleteAssoc needs to be moved into the in core system.
+
+As a work around,
+if you compile a non existent file, the apropriate files will be loaded,
+e.g.:
+\begin{verbatim}
+)co foo
+\end{verbatim}
+comment: this works in version 2
+<<*>>=
+)clear all
+
+--S 19
msq := Matrix SquareMatrix(2,POLY INT)
+--R
+--R
+--R (1) Matrix SquareMatrix(2,Polynomial Integer)
+--R Type:
Domain
+--E
+
+--S 20
m : msq := zero(2,2)
+--R
+--R ++0 0+ +0 0++
+--R || | | ||
+--R |+0 0+ +0 0+|
+--R (2) | |
+--R |+0 0+ +0 0+|
+--R || | | ||
+--R ++0 0+ +0 0++
+--R Type: Matrix SquareMatrix(2,Polynomial
Integer)
+--E
+
+--S 21
m(1,1) := matrix([[1,2],[a,b]])
+--R
+--R +1 2+
+--R (3) | |
+--R +a b+
+--R Type: SquareMatrix(2,Polynomial
Integer)
+--E
+
+--S 22
m(1,2) := matrix([[a,b],[2,b]])
+--R
+--R +a b+
+--R (4) | |
+--R +2 b+
+--R Type: SquareMatrix(2,Polynomial
Integer)
+--E
+
+--S 23
m(2,2) := matrix([[1,2],[2,b]])
+--R
+--R +1 2+
+--R (5) | |
+--R +2 b+
+--R Type: SquareMatrix(2,Polynomial
Integer)
+--E
+
+--S 24
m
+--R
+--R ++1 2+ +a b++
+--R || | | ||
+--R |+a b+ +2 b+|
+--R (6) | |
+--R |+0 0+ +1 2+|
+--R || | | ||
+--R ++0 0+ +2 b++
+--R Type: Matrix SquareMatrix(2,Polynomial
Integer)
+--E
+
+--S 25
m*m
+--R
+--R + + 2 ++
+--R |+2a + 1 2b + 2 + |2b + 2a + 4 b + 3b + 2a ||
+--R || | | ||
+--R || 2 | | 2 2 ||
+--R (7) |+a b + a b + 2a+ +4b + a + 2 2b + a b + 4+|
+--R | |
+--R | +0 0+ + 5 2b + 2+ |
+--R | | | | | |
+--R | +0 0+ | 2 | |
+--R + +2b + 2 b + 4+ +
+--R Type: Matrix SquareMatrix(2,Polynomial
Integer)
+--E
+
+--S 26
m**2
+--R
+--R + + 2 ++
+--R |+2a + 1 2b + 2 + |2b + 2a + 4 b + 3b + 2a ||
+--R || | | ||
+--R || 2 | | 2 2 ||
+--R (8) |+a b + a b + 2a+ +4b + a + 2 2b + a b + 4+|
+--R | |
+--R | +0 0+ + 5 2b + 2+ |
+--R | | | | | |
+--R | +0 0+ | 2 | |
+--R + +2b + 2 b + 4+ +
+--R Type: Matrix SquareMatrix(2,Polynomial
Integer)
+--E
+
+--S 27
m**3
+--R
+--R +matrix1 matrix2+
+--R (9) | |
+--R +matrix3 matrix4+
+--R
+--R
+--R
+--R + 2 +
+--R | 2a b + 4a + 1 2b + 2b + 4a + 2|
+--R where matrix1= | |
+--R | 2 2 3 |
+--R +a b + a b + 2a + a b + 4a b + 2a +
+--R
+--R and matrix2 =
+--R + 2 2 3 2
+
+--R | 2b + 12b + 2a + 7a + 8 b + 5b + (4a + 7)b + 4a + 8
|
+--R |
|
+--R | 2 2 2 3 2 2
|
+--R +6b + (a + 2a + 4)b + 2a + 4a + 10 3b + 2a b + (3a + 12)b + 2a
+ 4+
+--R
+--R +0 0+
+--R and matrix3= | |
+--R +0 0+
+--R
+--R + 2 +
+--R | 4b + 9 2b + 2b + 10|
+--R and matrix4= | |
+--R | 2 3 |
+--R +2b + 2b + 10 b + 8b + 4 +
+--R Type: Matrix SquareMatrix(2,Polynomial
Integer)
+--E
+
+--S 28
(m*m)*m
+--R
+--R +matrix1 matrix2+
+--R (10) | |
+--R +matrix3 matrix4+
+--R
+--R
+--R
+--R + 2 +
+--R | 2a b + 4a + 1 2b + 2b + 4a + 2|
+--R where matrix1= | |
+--R | 2 2 3 |
+--R +a b + a b + 2a + a b + 4a b + 2a +
+--R
+--R and matrix2 =
+--R + 2 2 3 2
+
+--R | 2b + 12b + 2a + 7a + 8 b + 5b + (4a + 7)b + 4a + 8
|
+--R |
|
+--R | 2 2 2 3 2 2
|
+--R +6b + (a + 2a + 4)b + 2a + 4a + 10 3b + 2a b + (3a + 12)b + 2a
+ 4+
+--R
+--R +0 0+
+--R and matrix3= | |
+--R +0 0+
+--R
+--R + 2 +
+--R | 4b + 9 2b + 2b + 10|
+--R and matrix4= | |
+--R | 2 3 |
+--R +2b + 2b + 10 b + 8b + 4 +
+--R Type: Matrix SquareMatrix(2,Polynomial
Integer)
+--E
+
+--S 29
mm:=m*m
+--R
+--R + + 2 ++
+--R |+2a + 1 2b + 2 + |2b + 2a + 4 b + 3b + 2a ||
+--R || | | ||
+--R || 2 | | 2 2 ||
+--R (11) |+a b + a b + 2a+ +4b + a + 2 2b + a b + 4+|
+--R | |
+--R | +0 0+ + 5 2b + 2+ |
+--R | | | | | |
+--R | +0 0+ | 2 | |
+--R + +2b + 2 b + 4+ +
+--R Type: Matrix SquareMatrix(2,Polynomial
Integer)
+--E
+
+--S 30
mm*m
+--R
+--R +matrix1 matrix2+
+--R (12) | |
+--R +matrix3 matrix4+
+--R
+--R
+--R
+--R + 2 +
+--R | 2a b + 4a + 1 2b + 2b + 4a + 2|
+--R where matrix1= | |
+--R | 2 2 3 |
+--R +a b + a b + 2a + a b + 4a b + 2a +
+--R
+--R and matrix2 =
+--R + 2 2 3 2
+
+--R | 2b + 12b + 2a + 7a + 8 b + 5b + (4a + 7)b + 4a + 8
|
+--R |
|
+--R | 2 2 2 3 2 2
|
+--R +6b + (a + 2a + 4)b + 2a + 4a + 10 3b + 2a b + (3a + 12)b + 2a
+ 4+
+--R
+--R +0 0+
+--R and matrix3= | |
+--R +0 0+
+--R
+--R + 2 +
+--R | 4b + 9 2b + 2b + 10|
+--R and matrix4= | |
+--R | 2 3 |
+--R +2b + 2b + 10 b + 8b + 4 +
+--R Type: Matrix SquareMatrix(2,Polynomial
Integer)
+--E
--------------------------------------------------------------------------
--- tpd/09/25/92 new equation.spad from johannes grabmeier
--------------------------------------------------------------------------
-
-
---Furthermore, we have function to put 0 or 1 on one side
---and factor the left hand side, after the right hand side
---is 0 and we have an IntegralDomain.
---Johannes
+@
+tpd/09/25/92 new equation.spad from johannes grabmeier
+Furthermore, we have function to put 0 or 1 on one side
+and factor the left hand side, after the right hand side
+is 0 and we have an IntegralDomain.\\
+Johannes
+<<*>>=
+)clear all
+--S 31
eq1 := (-6*x**3+13*x**2+4)=(-x**4+12*x)
---
---
--- 3 2 4
--- (1) - 6x + 13x + 4= - x + 12x
--- Type: Equation Polynomial Integer
--- Time: 1.61 (IN) + 1.47 (OT) = 3.08 sec
+--R
+--R 3 2 4
+--R (1) - 6x + 13x + 4= - x + 12x
+--R Type: Equation Polynomial
Integer
+--E
+
+--S 32
eq2 := x**4+13*x**2-12*x = 6*x**3-4
---
---
--- 4 2 3
--- (2) x + 13x - 12x= 6x - 4
--- Type: Equation Polynomial Integer
--- Time: 0.16 (IN) + 0.11 (OT) = 0.27 sec
+--R
+--R 4 2 3
+--R (2) x + 13x - 12x= 6x - 4
+--R Type: Equation Polynomial
Integer
+--E
+
+--S 33
eq := eq1*y**2+eq2
---
---
--- 3 2 2 4 2 4 2 3
--- (3) (- 6x + 13x + 4)y + x + 13x - 12x= (- x + 12x)y + 6x - 4
--- Type: Equation Polynomial Integer
--- Time: 0.26 (IN) + 0.01 (EV) + 0.01 (OT) + 1.54 (GC) = 1.82 sec
-swap %
---
---
--- 4 2 3 3 2 2 4 2
--- (4) (- x + 12x)y + 6x - 4= (- 6x + 13x + 4)y + x + 13x - 12x
--- Type: Equation Polynomial Integer
--- Time: 0.07 (OT) = 0.07 sec
-% + 4
---
---
--- 4 2 3 3 2 2 4 2
--- (5) (- x + 12x)y + 6x = (- 6x + 13x + 4)y + x + 13x - 12x + 4
--- Type: Equation Polynomial Integer
--- Time: 0.69 (IN) + 0.01 (OT) = 0.70 sec
-%-6*x**3
---
---
--- 4 2 3 2 2 4 3 2
--- (6) (- x + 12x)y = (- 6x + 13x + 4)y + x - 6x + 13x - 12x + 4
--- Type: Equation Polynomial Integer
--- Time: 0.19 (IN) + 0.01 (OT) = 0.20 sec
-leftZero %
---
---
--- 4 3 2 2 4 3 2
--- (7) 0= (x - 6x + 13x - 12x + 4)y + x - 6x + 13x - 12x + 4
--- Type: Equation Polynomial Integer
--- Time: 0.01 (IN) + 0.05 (OT) = 0.06 sec
-swap %
---
---
--- 4 3 2 2 4 3 2
--- (8) (x - 6x + 13x - 12x + 4)y + x - 6x + 13x - 12x + 4= 0
--- Type: Equation Polynomial Integer
--- Time: 0.01 (IN) = 0.01 sec
-factor lhs %
---
---
--- 2 2 2
--- (9) (x - 2) (x - 1) (y + 1)
--- Type: Factored Polynomial Integer
--- Time: 0.50 (IN) + 0.09 (EV) + 0.30 (OT) = 0.89 sec
-factorAndSplit eq
---
---
--- 2
--- (10) [x - 2= 0,x - 1= 0,y + 1= 0]
--- Type: List Equation Polynomial Integer
--- Time: 0.09 (EV) + 0.21 (OT) = 0.30 sec
-inv (eq :: EQ FRAC POLY INT)
---
---
--- 1 1
--- (11) - ------------------------------------= - ----------------------
--- 3 2 2 4 2 4 2 3
--- (6x - 13x - 4)y - x - 13x + 12x (x - 12x)y - 6x + 4
--- Type: Equation Fraction Polynomial Integer
--- Time: 0.03 (IN) = 0.03 sec
-- %
---
---
--- 1 1
--- (12) ------------------------------------= ----------------------
--- 3 2 2 4 2 4 2 3
--- (6x - 13x - 4)y - x - 13x + 12x (x - 12x)y - 6x + 4
--- Type: Equation Fraction Polynomial Integer
--- Time: 0.01 (IN) + 0.01 (OT) = 0.02 sec
-
-
--------------------------------------------------------------------
--- bmt/09/29/92 coercion bug
--- fails with:
--- failed cannot be coerced to mode
--- (Record (: coef1 (Integer)) (: coef2 (Integer)))
--- fixed on 09/29/92
--------------------------------------------------------------------
-)clear all
-(p,q):UP(x,INT)
-p:=3*x**4+11*x**2-4
-q:=9*x**4+9*x**2-4
+--R
+--R 3 2 2 4 2 4 2 3
+--R (3) (- 6x + 13x + 4)y + x + 13x - 12x= (- x + 12x)y + 6x - 4
+--R Type: Equation Polynomial
Integer
+--E
+
+--S 34
+t1:=swap eq
+--R
+--R 4 2 3 3 2 2 4 2
+--R (4) (- x + 12x)y + 6x - 4= (- 6x + 13x + 4)y + x + 13x - 12x
+--R Type: Equation Polynomial
Integer
+--E
+
+--S 35
+t2:=t1 + 4
+--R
+--R 4 2 3 3 2 2 4 2
+--R (5) (- x + 12x)y + 6x = (- 6x + 13x + 4)y + x + 13x - 12x + 4
+--R Type: Equation Polynomial
Integer
+--E
+
+--S 36
+t3:=t2-6*x**3
+--R
+--R 4 2 3 2 2 4 3 2
+--R (6) (- x + 12x)y = (- 6x + 13x + 4)y + x - 6x + 13x - 12x + 4
+--R Type: Equation Polynomial
Integer
+--E
+
+--S 37
+t4:=leftZero t3
+--R
+--R 4 3 2 2 4 3 2
+--R (7) 0= (x - 6x + 13x - 12x + 4)y + x - 6x + 13x - 12x + 4
+--R Type: Equation Polynomial
Integer
+--E
+
+--S 38
+t5:=swap t4
+--R
+--R 4 3 2 2 4 3 2
+--R (8) (x - 6x + 13x - 12x + 4)y + x - 6x + 13x - 12x + 4= 0
+--R Type: Equation Polynomial
Integer
+--E
+
+--S 39
+t6:=factor lhs t5
+--R
+--R 2 2 2
+--R (9) (x - 2) (x - 1) (y + 1)
+--R Type: Factored Polynomial
Integer
+--E
+
+--S 40
+t7:=factorAndSplit eq
+--R
+--R 2
+--R (10) [x - 2= 0,x - 1= 0,y + 1= 0]
+--R Type: List Equation Polynomial
Integer
+--E
+
+--S 41
+t8:=inv (eq :: EQ FRAC POLY INT)
+--R
+--R 1 1
+--R (11) - ------------------------------------= - ----------------------
+--R 3 2 2 4 2 4 2 3
+--R (6x - 13x - 4)y - x - 13x + 12x (x - 12x)y - 6x + 4
+--R Type: Equation Fraction Polynomial
Integer
+--E
+
+--S 42
+- t8
+--R
+--R 1 1
+--R (12) ------------------------------------= ----------------------
+--R 3 2 2 4 2 4 2 3
+--R (6x - 13x - 4)y - x - 13x + 12x (x - 12x)y - 6x + 4
+--R Type: Equation Fraction Polynomial
Integer
+--E
+
+@
+bmt/09/29/92 coercion bug\\
+fails with:
+\begin{verbatim}
+ failed cannot be coerced to mode
+ (Record (: coef1 (Integer)) (: coef2 (Integer)))
+\end{verbatim}
+fixed on 09/29/92
+
+Another subtle bug exists as of Nov 2008 release.
+The expression
+\begin{verbatim}
+ (p,q):UP(x,INT)
+\end{verbatim}
+generates the error
+\begin{verbatim}
+Daly Bug
+ Category, domain or package constructor : is not available.
+\end{verbatim}
+works fine at the top of this file but not here. Which means that
+some property is not being cleared properly. The symbol ``p'' does get
+properly declared despite the error message. --Tim
+<<*>>=
+)clear all
+
+--S 43
+(p1,p2):UP(x,INT)
+--R Type:
Void
+--E
+
+--S 44
+p1:=3*x**4+11*x**2-4
+--R
+--R 4 2
+--R (2) 3x + 11x - 4
+--R Type:
UnivariatePolynomial(x,Integer)
+--E
+
+--S 45
+p2:=9*x**4+9*x**2-4
+--R
+--R 4 2
+--R (3) 9x + 9x - 4
+--R Type:
UnivariatePolynomial(x,Integer)
+--E
+
+--S 46
myNextPrime: (INT,NNI) -> INT
+--R Type:
Void
+--E
+
+--S 47
myNextPrime(x,n)==nextPrime(x)$PRIMES(INT)
--- runs forever due to algebra bug in handling leading coefficients
-modularGcd([p,q])$InnerModularGcd(INT,UP(x,INT),67108859,myNextPrime)
+--R Type:
Void
+--E
--------------------------------------------------------------------
--- dewar/10/02/92
--- actually, this was never wrong
--------------------------------------------------------------------
+@
+Runs forever due to algebra bug in handling leading coefficients
+<<*>>=
+--S 48
+modularGcd([p1,p2])$InnerModularGcd(INT,UP(x,INT),67108859,myNextPrime)
+--R Compiling function myNextPrime with type (Integer,NonNegativeInteger
+--R ) -> Integer
+--R
+--R 2
+--R (6) 3x - 1
+--R Type:
UnivariatePolynomial(x,Integer)
+--E
+
+@
+dewar/10/02/92 actually, this was never wrong
+<<*>>=
+)clear all
+
+--S 49
numeric(%e ** %pi)
---gives
- (61)
--- 23.1406926327 7926900563 5493084578 4861294789 0532254433 2912200665
--- 6744971913 2534837468 7163222983 2730035
+--R
+--R (1) 23.1406926327 79269006
+--R Type:
Float
+--E
--------------------------------------------------------------------
--- themos/10/07/92
--------------------------------------------------------------------
+@
+themos/10/07/92
+<<*>>=
)clear all
+
+--S 50
y:=operator 'y
+--R
+--R (1) y
+--R Type:
BasicOperator
+--E
+
+--S 51
deqx:=differentiate(y(x),x,2)+differentiate(y(x),x)+y(x)
+--R
+--R ,, ,
+--R (2) y (x) + y (x) + y(x)
+--R
+--R Type: Expression
Integer
+--E
+
+--S 52
solve(deqx,y,x)
+--R
+--R x x
+--R +-+ - - - - +-+
+--R x\|3 2 2 x\|3
+--R (3) [particular= 0,basis= [cos(-----)%e ,%e sin(-----)]]
+--R 2 2
+--RType: Union(Record(particular: Expression Integer,basis: List Expression
Integer),...)
+--E
+
+--S 53
solve(deqx,y,x=0,[1])
+--R
+--R x
+--R +-+ - -
+--R x\|3 2
+--R (4) cos(-----)%e
+--R 2
+--R Type: Union(Expression
Integer,...)
+--E
+
+--S 54
deqt:=differentiate(y(t),t,2)+differentiate(y(t),t)+y(t)
+--R
+--R ,, ,
+--R (5) y (t) + y (t) + y(t)
+--R
+--R Type: Expression
Integer
+--E
+
+--S 55
solve(deqt,y,t)
+--R
+--R t t
+--R +-+ - - - - +-+
+--R t\|3 2 2 t\|3
+--R (6) [particular= 0,basis= [cos(-----)%e ,%e sin(-----)]]
+--R 2 2
+--RType: Union(Record(particular: Expression Integer,basis: List Expression
Integer),...)
+--E
+
+--S 56
solve(deqt,y,t=0,[1]) -- bug
+--R
+--R t
+--R +-+ - -
+--R t\|3 2
+--R (7) cos(-----)%e
+--R 2
+--R Type: Union(Expression
Integer,...)
+--E
+
+--S 57
deqz:=differentiate(y(z),z,2)+differentiate(y(z),z)+y(z)
+--R
+--R ,, ,
+--R (8) y (z) + y (z) + y(z)
+--R
+--R Type: Expression
Integer
+--E
+
+--S 58
solve(deqz,y,z)
+--R
+--R z z
+--R +-+ - - - - +-+
+--R z\|3 2 2 z\|3
+--R (9) [particular= 0,basis= [cos(-----)%e ,%e sin(-----)]]
+--R 2 2
+--RType: Union(Record(particular: Expression Integer,basis: List Expression
Integer),...)
+--E
+
+--S 59
solve(deqz,y,z=0,[1]) --bug
+--R
+--R z
+--R +-+ - -
+--R z\|3 2
+--R (10) cos(-----)%e
+--R 2
+--R Type: Union(Expression
Integer,...)
+--E
--------------------------------------------------------------------
--- themos/10/07/92 equation
--------------------------------------------------------------------
+@
+themos/10/07/92 equation
+<<*>>=
)clear all
+
+--S 60
y:=operator 'y
-deq:=D(y(x),x)+x**2=(y x)/x-(y x)**2
+--R
+--R (1) y
+--R Type:
BasicOperator
+--E
+--S 61
+deq:=D(y(x),x)+x**2=(y x)/x-(y x)**2
+--R
+--R 2
+--R , 2 - x y(x) + y(x)
+--R (2) y (x) + x = ----------------
+--R x
+--R Type: Equation Expression
Integer
+--E
--------------------------------------------------------------------
--- bmt/10/08/92 laplace
--------------------------------------------------------------------
+@
+bmt/10/08/92 laplace
+<<*>>=
)clear all
+
+--S 62
laplace(exp(-x**3)*x**7,x,s)
+--R
+--R 3
+--R 7 - x
+--R (1) laplace(x %e ,x,s)
+--R Type: Expression
Integer
+--E
+
+@
+bmt/09/30/92 nonlinear ODE
--------------------------------------------------------------------
--- bmt/09/30/92 nonlinear ODE
--------------------------------------------------------------------
--- Any hope of solving the following which appeared in Barry
--- Simon's computer algebra test suite:
--- (Apparently it has Bernoulli form)
+Any hope of solving the following which appeared in Barry
+Simon's computer algebra test suite:
+(Apparently it has Bernoulli form)
+<<*>>=
+)clear all
+
+--S 63
y:=operator 'y
+--R
+--R (1) y
+--R Type:
BasicOperator
+--E
+
+--S 64
x**2 * D(y x, x) + 2*x*(y x) - (y x)**3 = 0
+--R
+--R 2 , 3
+--R (2) x y (x) - y(x) + 2x y(x)= 0
+--R
+--R Type: Equation Expression
Integer
+--E
+
+--S 65
solve(%,y,x)
+--R
+--R 5 2
+--R (- 3x - 2)y(x) + 5x
+--R (3) ---------------------
+--R 5 2
+--R 5x y(x)
+--R Type: Union(Expression
Integer,...)
+--E
+@
+sit/10/08/92 factor bug
+<<*>>=
+)clear all
--------------------------------------------------------------------
--- sit/10/08/92 factor bug
--------------------------------------------------------------------
+--S 66
p:POLY FRAC INT := 3*(x+1)
--- is wrong (missing factor of 3).
+--R
+--R (1) 3x + 3
+--R Type: Polynomial Fraction
Integer
+--E
+
+@
+Is wrong (missing factor of 3).
+<<*>>=
+
+--S 67
factor(p)**2
+--R
+--R 2
+--R (2) 9(x + 1)
+--R Type: Factored Polynomial Fraction
Integer
+--E
+
+@
+henderson/10/08/92
+
+TextFile cliams that the operator readLineIfCan! returns ``failed'' when
+the end of the file is reached. In fact, I get a system error, which is
+not very useful:
+\begin{verbatim}
+(16) ->readLineIfCan!(fp)
+ >> System error:
+ Unexpected end of #<input stream "/home/mcd/work/work">.
+ You are being returned to the top level of the interpreter.
+\end{verbatim}
+
+<<*>>=
+)clear all
--------------------------------------------------------------------
--- henderson/10/08/92
--------------------------------------------------------------------
-)clear all
---TextFile cliams that the operator readLineIfCan! returns "failed" when
---the end of the file is reached. In fact, I get a system error, which is
---not very useful:
---(16) ->readLineIfCan!(fp)
--- >> System error:
--- Unexpected end of #<input stream "/home/mcd/work/work">.
--- You are being returned to the top level of the interpreter.
+--S 68
fout:TextFile:=open("/tmp/foo","output")
+--R
+--R (1) "/tmp/foo"
+--R Type:
TextFile
+--E
+
+--S 69
write!(fout,"foo")
+--R
+--R (2) "foo"
+--R Type:
String
+--E
+
+--S 70
close!(fout)
+--R
+--R (3) "/tmp/foo"
+--R Type:
TextFile
+--E
+
+--S 71
fin:TextFile:=open("/tmp/foo","input")
+--R
+--R (4) "/tmp/foo"
+--R Type:
TextFile
+--E
+
+--S 72
readLineIfCan!(fin)
+--R
+--R (5) "foo"
+--R Type:
Union(String,...)
+--E
+
+--S 73
readLineIfCan!(fin)
+--R
+--R (6) "failed"
+--R Type:
Union("failed",...)
+--E
+
+--S 74
close!(fin)
+--R
+--R (7) "/tmp/foo"
+--R Type:
TextFile
+--E
+
+--S 75
)lis (system "rm /tmp/foo")
+--R
+--RValue = 0
+--E
--------------------------------------------------------------------
--- bmt/10/08/92 factoring over SAEs
--------------------------------------------------------------------
+@
+bmt/10/08/92 factoring over SAEs
+<<*>>=
)clear all
+
+--S 76
a | a**2+1
-(x+a)*(x+a+1)
-factor %
+--R Your statement has resulted in the following assignments and
+--R declaration:
+--R
+--R SAEa := SimpleAlgebraicExtension(Fraction
Integer,UnivariatePolynomial(a,Fraction Integer),a*a+1)
+--R a : SAEa := a
+--R
+--R (1) a
+--RType: SimpleAlgebraicExtension(Fraction
Integer,UnivariatePolynomial(a,Fraction Integer),a*a+1)
+--E
+
+--S 77
+t1:=(x+a)*(x+a+1)
+--R
+--R 2
+--R (2) x + (2a + 1)x + a - 1
+--RType: Polynomial SimpleAlgebraicExtension(Fraction
Integer,UnivariatePolynomial(a,Fraction Integer),a*a+1)
+--E
+
+--S 78
+factor t1
+--R
+--R (3) (x + a + 1)(x + a)
+--RType: Factored Polynomial SimpleAlgebraicExtension(Fraction
Integer,UnivariatePolynomial(a,Fraction Integer),a*a+1)
+--E
--------------------------------------------------------------------
--- miller/10/09/92 local vars in types
--------------------------------------------------------------------
-)clear all
---Fixed (Victor's) bug with using local variables in type expressions
---assigned to local variables. It now gives a message and correctly uses
---interpret-only mode.
---The function changed was getAndEvalConstructorArgument from i-spec1.boot.
+@
+miller/10/09/92 local vars in types
+
+Fixed (Victor's) bug with using local variables in type expressions
+assigned to local variables. It now gives a message and correctly uses
+interpret-only mode.
+The function changed was getAndEvalConstructorArgument from i-spec1.boot.
+
+themos/10/19/92
--------------------------------------------------------------------
--- themos/10/19/92
--------------------------------------------------------------------
+Do this in a virgin system
+<<*>>=
)clear all
--- Do this in a virgin system
+
)set expose add constructor SquareMatrix
+
+--S 79
S2:= SquareMatrix(2,FRAC POLY INT);
-V2: S2 := matrix([[v,-v],[-v,v]]);
-I2: S2 := 1;
-m:=5;
+--R
+--R Type:
Domain
+--E
+
+--S 80
+V2: S2 := matrix([[v,-v],[-v,v]])
+--R
+--R + v - v+
+--R (2) | |
+--R +- v v +
+--R Type: SquareMatrix(2,Fraction Polynomial
Integer)
+--E
+
+--S 81
+I2: S2 := 1
+--R
+--R +1 0+
+--R (3) | |
+--R +0 1+
+--R Type: SquareMatrix(2,Fraction Polynomial
Integer)
+--E
+
+--S 82
+m:=5
+--R
+--R (4) 5
+--R Type:
PositiveInteger
+--E
+
+--S 83
l: List(S2) := append(cons(V2+h*I2,_
[(V2+2*h*I2) for i in 2 .. (m-1)]),_
[V2+h*I2])
+--R
+--R
+--R (5)
+--R +v + h - v + +v + 2h - v + +v + 2h - v + +v + 2h - v +
+--R [| |, | |, | |, | |,
+--R + - v v + h+ + - v v + 2h+ + - v v + 2h+ + - v v + 2h+
+--R +v + h - v +
+--R | |]
+--R + - v v + h+
+--R Type: List SquareMatrix(2,Fraction Polynomial
Integer)
+--E
+
+@
+Does this give an error of deleteAssoc not defined ??
+<<*>>=
+--S 84
A: SquareMatrix(m, S2) := diagonalMatrix(l)
--- did you get an error of deleteAssoc not defined ??
+--R
+--R +matrix1 matrix2 matrix2 matrix2 matrix2+
+--R | |
+--R |matrix2 matrix3 matrix2 matrix2 matrix2|
+--R | |
+--R (6) |matrix2 matrix2 matrix3 matrix2 matrix2|
+--R | |
+--R |matrix2 matrix2 matrix2 matrix3 matrix2|
+--R | |
+--R +matrix2 matrix2 matrix2 matrix2 matrix1+
+--R
+--R
+--R
+--R +v + h - v +
+--R where matrix1= | |
+--R + - v v + h+
+--R
+--R +0 0+
+--R and matrix2= | |
+--R +0 0+
+--R
+--R +v + 2h - v +
+--R and matrix3= | |
+--R + - v v + 2h+
+--R Type: SquareMatrix(5,SquareMatrix(2,Fraction Polynomial
Integer))
+--E
--- load definition of deleteAssoc
+@
+load definition of deleteAssoc
+\begin{verbatim}
)li (load "/spad/obj/rios/interp/c-util")
A: SquareMatrix(m, S2) := diagonalMatrix(l)
---now , can print.
+\end{verbatim}
+now , can print.
+
+bmt/10/19/92 squareFree bug
+
+Has an extra factor of 2.
+<<*>>=
+)clear all
--------------------------------------------------------------------
--- bmt/10/19/92 squareFree bug
--------------------------------------------------------------------
--- has an extra factor of 2.
+--S 85
squareFree((2*x*y+1)*(x*y+1)**2)
+--R
+--R 2
+--R (1) (x y + 1) (2x y + 1)
+--R Type: Factored Polynomial
Integer
+--E
+
+@
+bmt/10/19/92 division by zero error
+<<*>>=
+)clear all
--------------------------------------------------------------------
--- bmt/10/19/92 division by zero error
--------------------------------------------------------------------
+--S 86
limit(atan(1/sin(x)),x=0)
+--R
+--R %pi %pi
+--R (1) [leftHandLimit= - ---,rightHandLimit= ---]
+--R 2 2
+--RType: Union(Record(leftHandLimit: Union(OrderedCompletion Expression
Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression
Integer,"failed")),...)
+--E
+
+@
+bmt/10/19/92 limit problem
--------------------------------------------------------------------
--- bmt/10/19/92 limit problem
--------------------------------------------------------------------
+The left and right hand limits don't agree
+<<*>>=
)clear all
--- the left and right hand limits don't agree
+
+--S 87
limit(atan(-sin(x)/(cos(x)+e)),x=acos(-e))
--- the left and right hand limits don't agree
+--R
+--R %pi %pi
+--R (1) [leftHandLimit= - ---,rightHandLimit= ---]
+--R 2 2
+--RType: Union(Record(leftHandLimit: Union(OrderedCompletion Expression
Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression
Integer,"failed")),...)
+--E
+
+--S 88
limit(atan(1/(cos(x)+e)),x=acos(-e))
+--R
+--R %pi %pi
+--R (2) [leftHandLimit= ---,rightHandLimit= - ---]
+--R 2 2
+--RType: Union(Record(leftHandLimit: Union(OrderedCompletion Expression
Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression
Integer,"failed")),...)
+--E
--------------------------------------------------------------------
--- grabmeier/10/19/92 unable to compute the determinant of an 8x8
--------------------------------------------------------------------
+@
+grabmeier/10/19/92 unable to compute the determinant of an 8x8
+<<*>>=
)clear all
-D := MATRIX FRAC(POLY INT)
+--S 89
+D := MATRIX FRAC(POLY INT)
+--R
+--R (1) Matrix Fraction Polynomial Integer
+--R Type:
Domain
+--E
+--S 90
d : (INT, Boolean) -> POLY INT
+--R Type:
Void
+--E
+
+--S 91
d(i,ss) ==
ex := i rem 4
if ex < 0 then ex := ex+4
@@ -429,13 +1197,34 @@ d(i,ss) ==
ex = 1 => 'd
ex = 2 => 'dd
'ddd
-
-
+--R
+--R Type:
Void
+--E
-- 1,d,dd,ddd,s,sd,sdd,sddd
+--S 92
mTV4 : D := new(8,8,0)
-
+--R
+--R +0 0 0 0 0 0 0 0+
+--R | |
+--R |0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0|
+--R (4) | |
+--R |0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0|
+--R | |
+--R +0 0 0 0 0 0 0 0+
+--R Type: Matrix Fraction Polynomial
Integer
+--E
+
+--S 93
for i in 1..8 repeat
for j in 1..8 repeat
mTV4(i,j) :=
@@ -444,52 +1233,1022 @@ for i in 1..8 repeat
d(-i+j,true)
j <= 4 => d(i+j-2,true)
d(-i+j,false)
+--R
+--R Compiling function d with type (Integer,Boolean) -> Polynomial
+--R Integer
+--R Type:
Void
+--E
+
+--S 94
mTV4
+--R
+--R + 1 d dd ddd s sd sdd sddd+
+--R | |
+--R | d dd ddd 1 sddd s sd sdd |
+--R | |
+--R | dd ddd 1 d sdd sddd s sd |
+--R | |
+--R |ddd 1 d dd sd sdd sddd s |
+--R (6) | |
+--R | s sd sdd sddd 1 d dd ddd |
+--R | |
+--R | sd sdd sddd s ddd 1 d dd |
+--R | |
+--R |sdd sddd s sd dd ddd 1 d |
+--R | |
+--R +sddd s sd sdd d dd ddd 1 +
+--R Type: Matrix Fraction Polynomial
Integer
+--E
+
+--S 95
gdd4 := determinant mTV4
+--R
+--R (7)
+--R 8 2 2 2 2 6
+--R - sddd + (8s sdd + 4sd + 4ddd + 4dd + 4d + 4)sddd
+--R +
+--R 2
+--R - 8sd sdd + ((- 8dd - 8)ddd - 8d dd - 8d)sdd
+--R +
+--R 2
+--R (- 8s - 16d ddd - 16dd)sd + ((- 8dd - 8)ddd - 8d dd - 8d)s
+--R *
+--R 5
+--R sddd
+--R +
+--R 4 2 2 2 2 2
+--R 2sdd + (- 20s + 4ddd + 16d ddd + 4dd + 16dd + 4d + 4)sdd
+--R +
+--R 2
+--R - 8s sd + ((24dd + 24)ddd + 24d dd + 24d)sd
+--R +
+--R 2 2 2
+--R (- 16ddd + 16d ddd - 16dd + 16dd - 16d - 16)s
+--R *
+--R sdd
+--R +
+--R 4 2 2 2 2
+--R - 6sd + (- 4ddd - 4dd - 4d - 4)sd
+--R +
+--R 4
+--R ((24dd + 24)ddd + 24d dd + 24d)s sd + 2s
+--R +
+--R 2 2 2 2 4
+--R (4ddd + 16d ddd + 4dd + 16dd + 4d + 4)s - 6ddd
+--R +
+--R 2 2 2 2 4
+--R (- 8dd + 8dd - 4d - 8)ddd + (8d dd + 32d dd + 8d)ddd - 6dd
+--R +
+--R 2 2 2 4 2
+--R (- 8d - 4)dd + 8d dd - 6d - 8d - 6
+--R *
+--R 4
+--R sddd
+--R +
+--R 3
+--R (32s sd + (- 16dd - 16)ddd - 16d dd - 16d)sdd
+--R +
+--R 3
2
+--R (16sd + (- 32d ddd - 32dd)sd + ((16dd + 16)ddd + 16d dd +
16d)s)sdd
+--R +
+--R 2
+--R ((- 16dd - 16)ddd - 16d dd - 16d)sd
+--R +
+--R 3 2 2 2
+--R (32s + (- 32ddd - 32dd - 32d - 32)s)sd
+--R +
+--R 2 3
+--R ((16dd + 16)ddd + 16d dd + 16d)s + (16dd + 16)ddd
+--R +
+--R 2
+--R (- 16d dd - 16d)ddd
+--R +
+--R 3 2 2 2 3
+--R (16dd - 16dd + (- 16d - 16)dd - 16d + 16)ddd + 16d dd
+--R +
+--R 2 3 3
+--R - 16d dd + (16d - 16d)dd + 16d + 16d
+--R *
+--R sdd
+--R +
+--R 2 3 2
+--R (16s + 32d ddd + 32dd)sd + ((- 16dd - 16)ddd - 16d dd - 16d)s sd
+--R +
+--R 2 3 2 2
+--R (- 32d ddd - 32dd)s + 32d ddd + (- 16dd - 16)ddd
+--R +
+--R 3 3 2 2 2
+--R (- 64d dd + 32d )ddd + 32dd - 16d dd + 32dd - 16d
+--R *
+--R sd
+--R +
+--R 3
+--R ((- 16dd - 16)ddd - 16d dd - 16d)s
+--R +
+--R 3 2
+--R (16dd + 16)ddd + (- 16d dd - 16d)ddd
+--R +
+--R 3 2 2 2 3
+--R (16dd - 16dd + (- 16d - 16)dd - 16d + 16)ddd + 16d dd
+--R +
+--R 2 3 3
+--R - 16d dd + (16d - 16d)dd + 16d + 16d
+--R *
+--R s
+--R *
+--R 3
+--R sddd
+--R +
+--R 5 2 2 2 2 4
+--R - 8s sdd + (- 20sd + 4ddd + 16d ddd + 4dd + 16dd + 4d + 4)sdd
+--R +
+--R 3
3
+--R (((16dd + 16)ddd + 16d dd + 16d)sd + 16s + (- 32d ddd - 32dd)s)sdd
+--R +
+--R 2 2 2 2 2
+--R (- 56s + 24ddd - 32d ddd + 24dd - 32dd + 24d + 24)sd
+--R +
+--R ((- 16dd - 16)ddd - 16d dd - 16d)s sd
+--R +
+--R 2 2 2 2 4
3
+--R (24ddd - 32d ddd + 24dd - 32dd + 24d + 24)s - 8ddd - 16d
ddd
+--R +
+--R 2 2 2 3 4 3
+--R (- 8dd + 16d - 8)ddd + (96d dd - 16d )ddd - 8dd - 16dd
+--R +
+--R 2 2 4 2
+--R (- 8d + 16)dd - 16dd - 8d - 8d - 8
+--R *
+--R 2
+--R sdd
+--R +
+--R 4 3
+--R - 8s sd + ((- 16dd - 16)ddd - 16d dd - 16d)sd
+--R +
+--R 2 2 2 2
+--R (32ddd + 96d ddd + 32dd + 96dd + 32d + 32)s sd
+--R +
+--R 2 3
+--R ((- 16dd - 16)ddd - 16d dd - 16d)s + (- 16dd - 16)ddd
+--R +
+--R 2
+--R (16d dd + 16d)ddd
+--R +
+--R 3 2 2 2 3
+--R (- 16dd + 16dd + (16d + 16)dd + 16d - 16)ddd - 16d dd
+--R +
+--R 2 3 3
+--R 16d dd + (- 16d + 16d)dd - 16d - 16d
+--R *
+--R sd
+--R +
+--R 5 3
+--R - 8s + (- 32d ddd - 32dd)s
+--R +
+--R 4 2 2 2 4
+--R 8ddd + (- 32dd + 48d )ddd + (- 32d dd - 32d)ddd + 8dd
+--R +
+--R 2 2 4
+--R 48dd - 32d dd + 8d + 8
+--R *
+--R s
+--R *
+--R sdd
+--R +
+--R 6 2 2 2 4
+--R 4sd + (- 4ddd - 4dd - 4d - 4)sd
+--R +
+--R 3
+--R ((- 16dd - 16)ddd - 16d dd - 16d)s sd
+--R +
+--R 4 2 2 2 2
4
+--R - 20s + (24ddd - 32d ddd + 24dd - 32dd + 24d + 24)s - 4ddd
+--R +
+--R 2 2 2 2
4
+--R (16dd + 48dd - 88d + 16)ddd + (48d dd - 64d dd + 48d)ddd -
4dd
+--R +
+--R 2 2 2 4 2
+--R (16d - 88)dd + 48d dd - 4d + 16d - 4
+--R *
+--R 2
+--R sd
+--R +
+--R 3
+--R ((16dd + 16)ddd + 16d dd + 16d)s
+--R +
+--R 3 2
+--R (- 16dd - 16)ddd + (16d dd + 16d)ddd
+--R +
+--R 3 2 2 2 3
+--R (- 16dd + 16dd + (16d + 16)dd + 16d - 16)ddd - 16d dd
+--R +
+--R 2 3 3
+--R 16d dd + (- 16d + 16d)dd - 16d - 16d
+--R *
+--R s
+--R *
+--R sd
+--R +
+--R 2 2 2 4
+--R (4ddd + 16d ddd + 4dd + 16dd + 4d + 4)s
+--R +
+--R 4 3 2 2 2 3
+--R - 8ddd - 16d ddd + (- 8dd + 16d - 8)ddd + (96d dd - 16d
)ddd
+--R +
+--R 4 3 2 2 4 2
+--R - 8dd - 16dd + (- 8d + 16)dd - 16dd - 8d - 8d - 8
+--R *
+--R 2
+--R s
+--R +
+--R 6 2 2 4 3
+--R 4ddd + (4dd - 16dd - 4d + 4)ddd - 32d dd ddd
+--R +
+--R 4 2 2 2 4 2 2
+--R (4dd + (24d + 24)dd + 32d dd - 4d + 24d + 4)ddd
+--R +
+--R 4 3 2 3 6
+--R (- 16d dd - 32d dd + 32d dd + (- 32d - 32d)dd - 16d)ddd + 4dd
+--R +
+--R 2 4 4 2 2 4 6 4 2
+--R (4d - 4)dd + (4d + 24d - 4)dd - 16d dd + 4d + 4d + 4d + 4
+--R *
+--R 2
+--R sddd
+--R +
+--R 6 5
+--R 8sd sdd + ((- 8dd - 8)ddd - 8d dd - 8d)sdd
+--R +
+--R 2 2 2 2
+--R (- 8s - 16ddd + 16d ddd - 16dd + 16dd - 16d - 16)sd
+--R +
+--R ((24dd + 24)ddd + 24d dd + 24d)s
+--R *
+--R 4
+--R sdd
+--R +
+--R 3 2
+--R 32s sd + ((16dd + 16)ddd + 16d dd + 16d)sd
+--R +
+--R 2 2 2
+--R (- 32ddd - 32dd - 32d - 32)s sd
+--R +
+--R 2 3
+--R ((- 16dd - 16)ddd - 16d dd - 16d)s + (16dd + 16)ddd
+--R +
+--R 2
+--R (- 16d dd - 16d)ddd
+--R +
+--R 3 2 2 2 3
+--R (16dd - 16dd + (- 16d - 16)dd - 16d + 16)ddd + 16d dd
+--R +
+--R 2 3 3
+--R - 16d dd + (16d - 16d)dd + 16d + 16d
+--R *
+--R 3
+--R sdd
+--R +
+--R 5 3
+--R - 8sd + (- 32d ddd - 32dd)sd
+--R +
+--R 2
+--R ((- 16dd - 16)ddd - 16d dd - 16d)s sd
+--R +
+--R 4 2 2 2 2
4
+--R - 8s + (32ddd + 96d ddd + 32dd + 96dd + 32d + 32)s +
8ddd
+--R +
+--R 2 2 2 4 2
+--R (- 32dd + 48d )ddd + (- 32d dd - 32d)ddd + 8dd + 48dd
+--R +
+--R 2 4
+--R - 32d dd + 8d + 8
+--R *
+--R sd
+--R +
+--R 3
+--R ((- 16dd - 16)ddd - 16d dd - 16d)s
+--R +
+--R 3 2
+--R (- 16dd - 16)ddd + (16d dd + 16d)ddd
+--R +
+--R 3 2 2 2 3
+--R (- 16dd + 16dd + (16d + 16)dd + 16d - 16)ddd - 16d dd
+--R +
+--R 2 3 3
+--R 16d dd + (- 16d + 16d)dd - 16d - 16d
+--R *
+--R s
+--R *
+--R 2
+--R sdd
+--R +
+--R 4
+--R ((24dd + 24)ddd + 24d dd + 24d)sd
+--R +
+--R 3 2 2 2 3
+--R (32s + (- 32ddd - 32dd - 32d - 32)s)sd
+--R +
+--R 2 3
+--R ((- 16dd - 16)ddd - 16d dd - 16d)s + (- 16dd - 16)ddd
+--R +
+--R 2
+--R (16d dd + 16d)ddd
+--R +
+--R 3 2 2 2 3
+--R (- 16dd + 16dd + (16d + 16)dd + 16d - 16)ddd - 16d dd
+--R +
+--R 2 3 3
+--R 16d dd + (- 16d + 16d)dd - 16d - 16d
+--R *
+--R 2
+--R sd
+--R +
+--R 2 2 2 3
+--R (- 32ddd - 32dd - 32d - 32)s
+--R +
+--R 4 3 2 2 2
+--R 32ddd - 64d ddd + (96dd - 64d + 96)ddd
+--R +
+--R 3 4 3 2 2
+--R (- 128d dd - 64d )ddd + 32dd - 64dd + (96d - 64)dd
+--R +
+--R 4 2
+--R - 64dd + 32d + 96d + 32
+--R *
+--R s
+--R *
+--R sd
+--R +
+--R 4
+--R ((24dd + 24)ddd + 24d dd + 24d)s
+--R +
+--R 3 2
+--R (- 16dd - 16)ddd + (16d dd + 16d)ddd
+--R +
+--R 3 2 2 2 3
+--R (- 16dd + 16dd + (16d + 16)dd + 16d - 16)ddd - 16d dd
+--R +
+--R 2 3 3
+--R 16d dd + (- 16d + 16d)dd - 16d - 16d
+--R *
+--R 2
+--R s
+--R +
+--R 5 4
+--R (- 8dd - 8)ddd + (24d dd + 24d)ddd
+--R +
+--R 3 2 2 2 3
+--R (- 16dd + 16dd + (- 16d + 16)dd - 16d - 16)ddd
+--R +
+--R 3 2 3 3 2
+--R (16d dd - 16d dd + (- 16d - 16d)dd - 16d + 16d)ddd
+--R +
+--R 5 4 2 3 2 2
+--R - 8dd + 24dd + (16d - 16)dd + (- 16d - 16)dd
+--R +
+--R 4 2 4 2
+--R (24d - 16d + 24)dd + 24d + 16d - 8
+--R *
+--R ddd
+--R +
+--R 5 4 3 3 3 2
+--R - 8d dd + 24d dd + (- 16d - 16d)dd + (16d - 16d)dd
+--R +
+--R 5 3 5 3
+--R (- 8d + 16d + 24d)dd - 8d - 16d - 8d
+--R *
+--R sdd
+--R +
+--R 2 5 4
+--R (- 8s - 16d ddd - 16dd)sd + ((24dd + 24)ddd + 24d dd + 24d)s sd
+--R +
+--R 2 3 2 2
+--R (- 32d ddd - 32dd)s + 32d ddd + (- 16dd - 16)ddd
+--R +
+--R 3 3 2 2 2
+--R (- 64d dd + 32d )ddd + 32dd - 16d dd + 32dd - 16d
+--R *
+--R 3
+--R sd
+--R +
+--R 3
+--R ((16dd + 16)ddd + 16d dd + 16d)s
+--R +
+--R 3 2
+--R (- 16dd - 16)ddd + (16d dd + 16d)ddd
+--R +
+--R 3 2 2 2 3
+--R (- 16dd + 16dd + (16d + 16)dd + 16d - 16)ddd - 16d dd
+--R +
+--R 2 3 3
+--R 16d dd + (- 16d + 16d)dd - 16d - 16d
+--R *
+--R s
+--R *
+--R 2
+--R sd
+--R +
+--R 6 2 2 2 4
+--R 8s + (- 16ddd + 16d ddd - 16dd + 16dd - 16d - 16)s
+--R +
+--R 4 2 2 2 4
+--R 8ddd + (- 32dd + 48d )ddd + (- 32d dd - 32d)ddd + 8dd
+--R +
+--R 2 2 4
+--R 48dd - 32d dd + 8d + 8
+--R *
+--R 2
+--R s
+--R +
+--R 5 2 4 2 3
3
+--R - 16d ddd + (16dd + 16dd + 16)ddd + (- 32d dd + 32d -
32d)ddd
+--R +
+--R 4 3 2 2 2 2
2
+--R (16dd - 32dd + (- 32d - 32)dd + (96d - 32)dd - 32d +
16)ddd
+--R +
+--R 4 3 2 5 3 5
+--R (16d dd + (- 32d + 96d)dd - 16d - 32d + 16d)ddd - 16dd
+--R +
+--R 2 4 2 3 4 2 2
+--R 16d dd + (- 32d + 32)dd + (16d - 32d )dd
+--R +
+--R 4 2 4 2
+--R (16d - 32d - 16)dd + 16d + 16d
+--R *
+--R sd
+--R +
+--R 5
+--R ((- 8dd - 8)ddd - 8d dd - 8d)s
+--R +
+--R 3 2
+--R (16dd + 16)ddd + (- 16d dd - 16d)ddd
+--R +
+--R 3 2 2 2 3
+--R (16dd - 16dd + (- 16d - 16)dd - 16d + 16)ddd + 16d dd
+--R +
+--R 2 3 3
+--R - 16d dd + (16d - 16d)dd + 16d + 16d
+--R *
+--R 3
+--R s
+--R +
+--R 5 4
+--R (- 8dd - 8)ddd + (24d dd + 24d)ddd
+--R +
+--R 3 2 2 2 3
+--R (- 16dd + 16dd + (- 16d + 16)dd - 16d - 16)ddd
+--R +
+--R 3 2 3 3 2
+--R (16d dd - 16d dd + (- 16d - 16d)dd - 16d + 16d)ddd
+--R +
+--R 5 4 2 3 2 2
+--R - 8dd + 24dd + (16d - 16)dd + (- 16d - 16)dd
+--R +
+--R 4 2 4 2
+--R (24d - 16d + 24)dd + 24d + 16d - 8
+--R *
+--R ddd
+--R +
+--R 5 4 3 3 3 2
+--R - 8d dd + 24d dd + (- 16d - 16d)dd + (16d - 16d)dd
+--R +
+--R 5 3 5 3
+--R (- 8d + 16d + 24d)dd - 8d - 16d - 8d
+--R *
+--R s
+--R *
+--R sddd
+--R +
+--R 8 2 2 2 2 6
+--R - sdd + (4s + 4ddd + 4dd + 4d + 4)sdd
+--R +
+--R 2 5
+--R (- 8s sd + ((- 8dd - 8)ddd - 8d dd - 8d)sd + (- 16d ddd - 16dd)s)sdd
+--R +
+--R 4 2 2 2 2
+--R 2sd + (4ddd + 16d ddd + 4dd + 16dd + 4d + 4)sd
+--R +
+--R 4
+--R ((24dd + 24)ddd + 24d dd + 24d)s sd - 6s
+--R +
+--R 2 2 2 2 4 2 2 2
+--R (- 4ddd - 4dd - 4d - 4)s - 6ddd + (- 8dd + 8dd - 4d - 8)ddd
+--R +
+--R 2 4 2 2 2 4 2
+--R (8d dd + 32d dd + 8d)ddd - 6dd + (- 8d - 4)dd + 8d dd - 6d - 8d
- 6
+--R *
+--R 4
+--R sdd
+--R +
+--R 3 3
2
+--R ((- 16dd - 16)ddd - 16d dd - 16d)sd + (16s + (- 32d ddd -
32dd)s)sd
+--R +
+--R 2 3
+--R ((- 16dd - 16)ddd - 16d dd - 16d)s + (16dd + 16)ddd
+--R +
+--R 2
+--R (- 16d dd - 16d)ddd
+--R +
+--R 3 2 2 2 3
+--R (16dd - 16dd + (- 16d - 16)dd - 16d + 16)ddd + 16d dd
+--R +
+--R 2 3 3
+--R - 16d dd + (16d - 16d)dd + 16d + 16d
+--R *
+--R sd
+--R +
+--R 3
+--R (32d ddd + 32dd)s
+--R +
+--R 3 2 2 3 3
+--R 32d ddd + (- 16dd - 16)ddd + (- 64d dd + 32d )ddd + 32dd
+--R +
+--R 2 2 2
+--R - 16d dd + 32dd - 16d
+--R *
+--R s
+--R *
+--R 3
+--R sdd
+--R +
+--R 2 2 2 2 4
+--R (- 20s + 4ddd + 16d ddd + 4dd + 16dd + 4d + 4)sd
+--R +
+--R 3
+--R ((16dd + 16)ddd + 16d dd + 16d)s sd
+--R +
+--R 2 2 2 2 4
3
+--R (24ddd - 32d ddd + 24dd - 32dd + 24d + 24)s - 8ddd - 16d
ddd
+--R +
+--R 2 2 2 3 4 3
+--R (- 8dd + 16d - 8)ddd + (96d dd - 16d )ddd - 8dd - 16dd
+--R +
+--R 2 2 4 2
+--R (- 8d + 16)dd - 16dd - 8d - 8d - 8
+--R *
+--R 2
+--R sd
+--R +
+--R 3
+--R ((- 16dd - 16)ddd - 16d dd - 16d)s
+--R +
+--R 3 2
+--R (- 16dd - 16)ddd + (16d dd + 16d)ddd
+--R +
+--R 3 2 2 2 3
+--R (- 16dd + 16dd + (16d + 16)dd + 16d - 16)ddd - 16d dd
+--R +
+--R 2 3 3
+--R 16d dd + (- 16d + 16d)dd - 16d - 16d
+--R *
+--R s
+--R *
+--R sd
+--R +
+--R 6 2 2 2 4
+--R 4s + (- 4ddd - 4dd - 4d - 4)s
+--R +
+--R 4 2 2 2
+--R - 4ddd + (16dd + 48dd - 88d + 16)ddd
+--R +
+--R 2 4 2 2 2
4
+--R (48d dd - 64d dd + 48d)ddd - 4dd + (16d - 88)dd + 48d dd -
4d
+--R +
+--R 2
+--R 16d - 4
+--R *
+--R 2
+--R s
+--R +
+--R 6 2 2 4 3
+--R 4ddd + (4dd - 16dd - 4d + 4)ddd - 32d dd ddd
+--R +
+--R 4 2 2 2 4 2 2
+--R (4dd + (24d + 24)dd + 32d dd - 4d + 24d + 4)ddd
+--R +
+--R 4 3 2 3 6
+--R (- 16d dd - 32d dd + 32d dd + (- 32d - 32d)dd - 16d)ddd + 4dd
+--R +
+--R 2 4 4 2 2 4 6 4 2
+--R (4d - 4)dd + (4d + 24d - 4)dd - 16d dd + 4d + 4d + 4d + 4
+--R *
+--R 2
+--R sdd
+--R +
+--R 6 5
+--R 8s sd + ((- 8dd - 8)ddd - 8d dd - 8d)sd
+--R +
+--R 2 2 2 4
+--R (- 16ddd + 16d ddd - 16dd + 16dd - 16d - 16)s sd
+--R +
+--R 2 3
+--R ((16dd + 16)ddd + 16d dd + 16d)s + (16dd + 16)ddd
+--R +
+--R 2
+--R (- 16d dd - 16d)ddd
+--R +
+--R 3 2 2 2 3
+--R (16dd - 16dd + (- 16d - 16)dd - 16d + 16)ddd + 16d dd
+--R +
+--R 2 3 3
+--R - 16d dd + (16d - 16d)dd + 16d + 16d
+--R *
+--R 3
+--R sd
+--R +
+--R 5 3
+--R - 8s + (- 32d ddd - 32dd)s
+--R +
+--R 4 2 2 2 4
+--R 8ddd + (- 32dd + 48d )ddd + (- 32d dd - 32d)ddd + 8dd
+--R +
+--R 2 2 4
+--R 48dd - 32d dd + 8d + 8
+--R *
+--R s
+--R *
+--R 2
+--R sd
+--R +
+--R 4
+--R ((24dd + 24)ddd + 24d dd + 24d)s
+--R +
+--R 3 2
+--R (- 16dd - 16)ddd + (16d dd + 16d)ddd
+--R +
+--R 3 2 2 2 3
+--R (- 16dd + 16dd + (16d + 16)dd + 16d - 16)ddd - 16d dd
+--R +
+--R 2 3 3
+--R 16d dd + (- 16d + 16d)dd - 16d - 16d
+--R *
+--R 2
+--R s
+--R +
+--R 5 4
+--R (- 8dd - 8)ddd + (24d dd + 24d)ddd
+--R +
+--R 3 2 2 2 3
+--R (- 16dd + 16dd + (- 16d + 16)dd - 16d - 16)ddd
+--R +
+--R 3 2 3 3 2
+--R (16d dd - 16d dd + (- 16d - 16d)dd - 16d + 16d)ddd
+--R +
+--R 5 4 2 3 2 2
+--R - 8dd + 24dd + (16d - 16)dd + (- 16d - 16)dd
+--R +
+--R 4 2 4 2
+--R (24d - 16d + 24)dd + 24d + 16d - 8
+--R *
+--R ddd
+--R +
+--R 5 4 3 3 3 2
+--R - 8d dd + 24d dd + (- 16d - 16d)dd + (16d - 16d)dd
+--R +
+--R 5 3 5 3
+--R (- 8d + 16d + 24d)dd - 8d - 16d - 8d
+--R *
+--R sd
+--R +
+--R 5
+--R (- 16d ddd - 16dd)s
+--R +
+--R 3 2 2 3 3
+--R 32d ddd + (- 16dd - 16)ddd + (- 64d dd + 32d )ddd + 32dd
+--R +
+--R 2 2 2
+--R - 16d dd + 32dd - 16d
+--R *
+--R 3
+--R s
+--R +
+--R 5 2 4 2 3
3
+--R - 16d ddd + (16dd + 16dd + 16)ddd + (- 32d dd + 32d -
32d)ddd
+--R +
+--R 4 3 2 2 2 2
2
+--R (16dd - 32dd + (- 32d - 32)dd + (96d - 32)dd - 32d +
16)ddd
+--R +
+--R 4 3 2 5 3 5
+--R (16d dd + (- 32d + 96d)dd - 16d - 32d + 16d)ddd - 16dd
+--R +
+--R 2 4 2 3 4 2 2
+--R 16d dd + (- 32d + 32)dd + (16d - 32d )dd
+--R +
+--R 4 2 4 2
+--R (16d - 32d - 16)dd + 16d + 16d
+--R *
+--R s
+--R *
+--R sdd
+--R +
+--R 8 2 2 2 6
5
+--R - sd + (4ddd + 4dd + 4d + 4)sd + ((- 8dd - 8)ddd - 8d dd - 8d)s sd
+--R +
+--R 4 2 2 2 2 4
+--R 2s + (4ddd + 16d ddd + 4dd + 16dd + 4d + 4)s - 6ddd
+--R +
+--R 2 2 2 2 4
+--R (- 8dd + 8dd - 4d - 8)ddd + (8d dd + 32d dd + 8d)ddd - 6dd
+--R +
+--R 2 2 2 4 2
+--R (- 8d - 4)dd + 8d dd - 6d - 8d - 6
+--R *
+--R 4
+--R sd
+--R +
+--R 3
+--R ((- 16dd - 16)ddd - 16d dd - 16d)s
+--R +
+--R 3 2
+--R (16dd + 16)ddd + (- 16d dd - 16d)ddd
+--R +
+--R 3 2 2 2 3
+--R (16dd - 16dd + (- 16d - 16)dd - 16d + 16)ddd + 16d dd
+--R +
+--R 2 3 3
+--R - 16d dd + (16d - 16d)dd + 16d + 16d
+--R *
+--R s
+--R *
+--R 3
+--R sd
+--R +
+--R 2 2 2 4
+--R (4ddd + 16d ddd + 4dd + 16dd + 4d + 4)s
+--R +
+--R 4 3 2 2 2 3
+--R - 8ddd - 16d ddd + (- 8dd + 16d - 8)ddd + (96d dd - 16d
)ddd
+--R +
+--R 4 3 2 2 4 2
+--R - 8dd - 16dd + (- 8d + 16)dd - 16dd - 8d - 8d - 8
+--R *
+--R 2
+--R s
+--R +
+--R 6 2 2 4 3
+--R 4ddd + (4dd - 16dd - 4d + 4)ddd - 32d dd ddd
+--R +
+--R 4 2 2 2 4 2 2
+--R (4dd + (24d + 24)dd + 32d dd - 4d + 24d + 4)ddd
+--R +
+--R 4 3 2 3 6
+--R (- 16d dd - 32d dd + 32d dd + (- 32d - 32d)dd - 16d)ddd + 4dd
+--R +
+--R 2 4 4 2 2 4 6 4 2
+--R (4d - 4)dd + (4d + 24d - 4)dd - 16d dd + 4d + 4d + 4d + 4
+--R *
+--R 2
+--R sd
+--R +
+--R 5
+--R ((- 8dd - 8)ddd - 8d dd - 8d)s
+--R +
+--R 3 2
+--R (16dd + 16)ddd + (- 16d dd - 16d)ddd
+--R +
+--R 3 2 2 2 3
+--R (16dd - 16dd + (- 16d - 16)dd - 16d + 16)ddd + 16d dd
+--R +
+--R 2 3 3
+--R - 16d dd + (16d - 16d)dd + 16d + 16d
+--R *
+--R 3
+--R s
+--R +
+--R 5 4
+--R (- 8dd - 8)ddd + (24d dd + 24d)ddd
+--R +
+--R 3 2 2 2 3
+--R (- 16dd + 16dd + (- 16d + 16)dd - 16d - 16)ddd
+--R +
+--R 3 2 3 3 2
+--R (16d dd - 16d dd + (- 16d - 16d)dd - 16d + 16d)ddd
+--R +
+--R 5 4 2 3 2 2
+--R - 8dd + 24dd + (16d - 16)dd + (- 16d - 16)dd
+--R +
+--R 4 2 4 2
+--R (24d - 16d + 24)dd + 24d + 16d - 8
+--R *
+--R ddd
+--R +
+--R 5 4 3 3 3 2
+--R - 8d dd + 24d dd + (- 16d - 16d)dd + (16d - 16d)dd
+--R +
+--R 5 3 5 3
+--R (- 8d + 16d + 24d)dd - 8d - 16d - 8d
+--R *
+--R s
+--R *
+--R sd
+--R +
+--R 8 2 2 2 6
+--R - s + (4ddd + 4dd + 4d + 4)s
+--R +
+--R 4 2 2 2 2
+--R - 6ddd + (- 8dd + 8dd - 4d - 8)ddd + (8d dd + 32d dd + 8d)ddd
+--R +
+--R 4 2 2 2 4 2
+--R - 6dd + (- 8d - 4)dd + 8d dd - 6d - 8d - 6
+--R *
+--R 4
+--R s
+--R +
+--R 6 2 2 4 3
+--R 4ddd + (4dd - 16dd - 4d + 4)ddd - 32d dd ddd
+--R +
+--R 4 2 2 2 4 2 2
+--R (4dd + (24d + 24)dd + 32d dd - 4d + 24d + 4)ddd
+--R +
+--R 4 3 2 3 6
+--R (- 16d dd - 32d dd + 32d dd + (- 32d - 32d)dd - 16d)ddd + 4dd
+--R +
+--R 2 4 4 2 2 4 6 4 2
+--R (4d - 4)dd + (4d + 24d - 4)dd - 16d dd + 4d + 4d + 4d + 4
+--R *
+--R 2
+--R s
+--R +
+--R 8 2 6 2 5
+--R - ddd + (8dd + 4d )ddd + (- 8d dd - 8d)ddd
+--R +
+--R 4 2 2 4 4
+--R (2dd - 20dd - 8d dd - 6d + 2)ddd
+--R +
+--R 3 3 2 3 3
+--R (32d dd + 16d dd + 32d dd + 16d )ddd
+--R +
+--R 5 2 4 3 2 2 4 6 2 2
+--R (- 8dd - 20d dd + 16dd - 56d dd + (- 8d - 8)dd + 4d - 20d )ddd
+--R +
+--R 6 4 3 3 5 2 3 5
8
+--R (8d dd - 8d dd + 32d dd + (- 8d - 8d)dd + 32d dd - 8d + 8d)ddd -
dd
+--R +
+--R 6 2 5 4 4 2 3 4 2 6 2
+--R 4dd - 8d dd + (2d - 6)dd + 16d dd + (- 20d + 4)dd + (8d - 8d
)dd
+--R +
+--R 8 4
+--R - d + 2d - 1
+--R Type: Fraction Polynomial
Integer
+--E
--------------------------------------------------------------------
--- sutor/09/28/92 operator bug?
--------------------------------------------------------------------
+@
+sutor/09/28/92 operator bug?
+<<*>>=
)clear all
+
+--S 96
L n ==
n = 0 => 1
n = 1 => x
(2*n-1)/n * x * L(n-1) - (n-1)/n * L(n-2)
+--R
+--R Type:
Void
+--E
+
+--S 97
dx:=operator("D")::OP(POLY FRAC INT)
+--R
+--R (2) D
+--R Type: Operator Polynomial Fraction
Integer
+--E
+
+--S 98
evaluate(dx,p+-> differentiate(p,'x))
+--R
+--R There are 4 exposed and 1 unexposed library operations named
+--R evaluate having 2 argument(s) but none was determined to be
+--R applicable. Use HyperDoc Browse, or issue
+--R )display op evaluate
+--R to learn more about the available operations. Perhaps
+--R package-calling the operation or using coercions on the arguments
+--R will allow you to apply the operation.
+--R
+--RDaly Bug
+--R Cannot find a definition or applicable library operation named
+--R evaluate with argument type(s)
+--R Operator Polynomial Fraction Integer
+--R AnonymousFunction
+--R
+--R Perhaps you should use "@" to indicate the required return type,
+--R or "$" to specify which version of the function you need.
+--E
+
+--S 99
E n == (1-x**2)*dx**2-2*x*dx+n*(n+1)
+--R Type:
Void
+--E
+
+--S 100
L 15
--- used to fail on this line
+--R Compiling function L with type Integer -> Polynomial Fraction
+--R Integer
+--R Compiling function L as a recurrence relation.
+--R
+--R (4)
+--R 9694845 15 35102025 13 50702925 11 37182145 9 14549535 7
+--R ------- x - -------- x + -------- x - -------- x + -------- x
+--R 2048 2048 2048 2048 2048
+--R +
+--R 2909907 5 255255 3 6435
+--R - ------- x + ------ x - ---- x
+--R 2048 2048 2048
+--R Type: Polynomial Fraction
Integer
+--E
+
+@
+Used to fail on this line
+<<*>>=
+
+--S 101
E 15
+--R Compiling function E with type PositiveInteger -> Operator
+--R Polynomial Fraction Integer
+--R
+--R 2 2
+--R (5) 240 - 2x D - (x - 1)D
+--R Type: Operator Polynomial Fraction
Integer
+--E
--------------------------------------------------------------------
--- bmt/10/12/92 EFSTRUC recursion problem
--------------------------------------------------------------------
+@
+bmt/10/12/92 EFSTRUC recursion problem
+<<*>>=
)clear all
+
+--S 102
bug:=(1+x**(1/4))**(1/3)/(x**(1/2))
--- gives a value stack overflow
-integrate(bug,x) -- seems to go into infinite recursion in rischNormalize
+--R
+--R +--------+
+--R 3|4+-+
+--R \|\|x + 1
+--R (1) -----------
+--R +-+
+--R \|x
+--R Type: Expression
Integer
+--E
--------------------------------------------------------------------
--- james/10/28/92 coerce bug
--------------------------------------------------------------------
+@
+Gives a value stack overflow
+
+Seems to go into infinite recursion in rischNormalize
+<<*>>=
+--S 103
+integrate(bug,x)
+--R
+--R +--------+
+--R 4+-+2 4+-+ 3|4+-+
+--R (12\|x + 3\|x - 9)\|\|x + 1
+--R (2) --------------------------------
+--R 7
+--R Type: Union(Expression
Integer,...)
+--E
+
+@
+james/10/28/92 coerce bug
+
+Improper character in Roman numeral: G
+<<*>>=
)clear all
--- improper character in Roman numeral: G
+
+--S 104
g::ROMAN
+--R
+--R
+--RDaly Bug
+--R >> Error detected within library code:
+--R Improper character in Roman numeral:
+--R #\G
+--R
+--R Continuing to read the file...
+--R
+--E
+
+@
+grabm/10/28/92 runs forever
--------------------------------------------------------------------
--- grabm/10/28/92 runs forever
--------------------------------------------------------------------
+factor is quick but sqrt runs forever
+<<*>>=
)clear all
--- this is quick
+
+--S 105
factor 1068303355883998767544567663620885466990173600
--- but this runs forever
-sqrt 1068303355883998767544567663620885466990173600
+--R
+--R 5 7 2 4 2 2 2
+--R (1) 2 3 5 7 17 19 23 3343 4219 326705949951846198203
+--R Type: Factored
Integer
+--E
+--S 106
+sqrt 1068303355883998767544567663620885466990173600
+--R
+--R +-----------------------------+
+--R (2) 196571340\|27647393656301898872761810506
+--R Type:
AlgebraicNumber
+--E
--------------------------------------------------------------------
--- themos/11/05/92 fortran output bug
--------------------------------------------------------------------
+@
+themos/11/05/92 fortran output bug
+<<*>>=
)clear all
-- REAL T7,T6,T5,T4,T3,T2,T1
-- T1=x*x
@@ -503,670 +2262,3465 @@ sqrt 1068303355883998767544567663620885466990173600
-- &16.*x)*T4+(-T5+16.*T6)*y)*COS(x))/(y**8+4.*T1*y**6+6.*T7*y**4+4.*x
-- &**6*y*y+x**8)
-- T7 is referenced before it is defined
-a1:=sin(x)/(x**2+y**2);a2:=D(a1,[x,y]);a2+D(a2,x)
-
--------------------------------------------------------------------
--- grabm/11/05/92
--- comment:
---I believe the "bug" is that the interpreter is converting gb which is
---a list of dmp style polynomials to a list of POLY type polynomials.
---Unfortunately the result of such a conversion is no longer a groebner basis.
---I don't see any reasonable way to prevent this, and would therefore have to
---classify this as "user error", when working with grobner bases, the user
---needs to force all his polynomials to be the appropriate type (thus having
---the correct ordering).
--------------------------------------------------------------------
--- expressing a symmetric functions in terms of elementary symm. functions
--- working in POLY INT is o.k.
+
+--S 107
+a1:=sin(x)/(x**2+y**2)
+--R
+--R sin(x)
+--R (1) -------
+--R 2 2
+--R y + x
+--R Type: Expression
Integer
+--E
+
+--S 108
+a2:=D(a1,[x,y])
+--R
+--R 3 2
+--R 8x y sin(x) + (- 2y - 2x y)cos(x)
+--R (2) ----------------------------------
+--R 6 2 4 4 2 6
+--R y + 3x y + 3x y + x
+--R Type: Expression
Integer
+--E
+
+--S 109
+a2+D(a2,x)
+--R
+--R (3)
+--R 5 2 3 4 3 2
+--R (2y + (4x + 8x + 8)y + (2x + 8x - 40x )y)sin(x)
+--R +
+--R 5 2 3 4 3
+--R (- 2y + (- 4x + 16x)y + (- 2x + 16x )y)cos(x)
+--R /
+--R 8 2 6 4 4 6 2 8
+--R y + 4x y + 6x y + 4x y + x
+--R Type: Expression
Integer
+--E
+
+@
+grabm/11/05/92
+
+I believe the ``bug'' is that the interpreter is converting gb which is
+a list of dmp style polynomials to a list of POLY type polynomials.
+Unfortunately the result of such a conversion is no longer a groebner basis.
+I don't see any reasonable way to prevent this, and would therefore have to
+classify this as ``user error'', when working with grobner bases, the user
+needs to force all his polynomials to be the appropriate type (thus having
+the correct ordering).
+
+Expressing a symmetric functions in terms of elementary symm. functions
+working in POLY INT is o.k.
+<<*>>=
+)clear all
+
+--S 110
R := FRAC INT
--- (1) Fraction Integer
+--R
+--R (1) Fraction Integer
+--R Type:
Domain
+--E
+
+--S 111
lpx : List POLY R := [x1+x2+x3, x1*x2+x1*x3+x2*x3,x1*x2*x3]
--- (2) [x3 + x2 + x1,(x2 + x1)x3 + x1 x2,x1 x2 x3]
-lip := [lpx.1-e3, lpx.2-e2, lpx.3-e1]
--- (3) [x3 + x2 + x1 - e3,(x2 + x1)x3 + x1 x2 - e2,x1 x2 x3 - e1]
+--R
+--R (2) [x3 + x2 + x1,(x2 + x1)x3 + x1 x2,x1 x2 x3]
+--R Type: List Polynomial Fraction
Integer
+--E
+
+--S 112
+lip := [lpx.1-e3, lpx.2-e2, lpx.3-e1]--R
+--R
+--R (3) [x3 + x2 + x1 - e3,(x2 + x1)x3 + x1 x2 - e2,x1 x2 x3 - e1]
+--R Type: List Polynomial Fraction
Integer
+--E
+
+--S 113
gbp := groebner lip
--- (4)
--- 2 2
--- [x3 + x2 + x1 - e3, x2 + (x1 - e3)x2 + x1 - e3 x1 + e2,
--- 3 2
--- x1 - e3 x1 + e2 x1 - e1]
+--R
+--R (4)
+--R 2 2
+--R [x3 + x2 + x1 - e3, x2 + (x1 - e3)x2 + x1 - e3 x1 + e2,
+--R 3 2
+--R x1 - e3 x1 + e2 x1 - e1]
+--R Type: List Polynomial Fraction
Integer
+--E
+
+--S 114
normalForm(x1**2+x2**2+x3**2,gbp)
--- 2
--- (5) e3 - 2e2
--- working in DMP is o.k.
+--R
+--R 2
+--R (5) e3 - 2e2
+--R Type: Polynomial Fraction
Integer
+--E
+
+--S 115
dmp := DMP([x1,x2,x3,e1,e2,e3],INT)
--- (6) DistributedMultivariatePolynomial([x1,x2,x3,e1,e2,e3],Integer)
+--R
+--R (6) DistributedMultivariatePolynomial([x1,x2,x3,e1,e2,e3],Integer)
+--R Type:
Domain
+--E
+
+--S 116
li : List dmp := [lpx.1-e1, lpx.2-e2, lpx.3-e3]
--- (7) [x1 + x2 + x3 - e1,x1 x2 + x1 x3 + x2 x3 - e2,x1 x2 x3 - e3]
+--R
+--R (7) [x1 + x2 + x3 - e1,x1 x2 + x1 x3 + x2 x3 - e2,x1 x2 x3 - e3]
+--R Type: List
DistributedMultivariatePolynomial([x1,x2,x3,e1,e2,e3],Integer)
+--E
+
+--S 117
gb := groebner li
--- (8)
--- 2 2
--- [x1 + x2 + x3 - e1, x2 + x2 x3 - x2 e1 + x3 - x3 e1 + e2,
--- 3 2
--- x3 - x3 e1 + x3 e2 - e3]
+--R
+--R (8)
+--R 2 2
+--R [x1 + x2 + x3 - e1, x2 + x2 x3 - x2 e1 + x3 - x3 e1 + e2,
+--R 3 2
+--R x3 - x3 e1 + x3 e2 - e3]
+--R Type: List
DistributedMultivariatePolynomial([x1,x2,x3,e1,e2,e3],Integer)
+--E
+
+--S 118
p:dmp:=(x1**2+x2**2+x3**2)
--- 2 2 2
--- (9) x1 + x2 + x3
+--R
+--R 2 2 2
+--R (9) x1 + x2 + x3
+--R Type:
DistributedMultivariatePolynomial([x1,x2,x3,e1,e2,e3],Integer)
+--E
+
+--S 119
normalForm(p,gb)
--- 2
--- (10) e1 - 2e2
--- but we have some problems (coercion? ordering?) here:
+--R
+--R 2
+--R (10) e1 - 2e2
+--RType: DistributedMultivariatePolynomial([x1,x2,x3,e1,e2,e3],Fraction
Integer)
+--E
+
+--S 120
normalForm(x1**2+x2**2+x3**2,gb)
--- 2 2 2
--- (11) 2x2 + (2x1 - 2e1)x2 + 2x1 - 2e1 x1 + e1
+--R
+--R 2 2 2
+--R (11) 2x2 + (2x1 - 2e1)x2 + 2x1 - 2e1 x1 + e1
+--R Type: Polynomial Fraction
Integer
+--E
+@
+bmt/11/17/92 Ifintegrate
+
+It seems that one cannot simply parametrize reducible curves.
+Any attempt at parametrization can only follow one branch.
+For curves of the form $y^2=a^2x^2+bx+c$, these are only reducible
+if $a^2x^2+bx+c$ is a perfect square. Basically I think we need to
+choose 1 branch in this case. Under some circumstances we could
+try to proceed down both branches and then unify the result, but for
+the time being, perhaps we should just perform the substitution
+$y=sqrt(a^2x^2+bx+c)$ whenever that is a perfect square.
+(then we don't even need to do any back subsitution or change to dx).
+
+For another example, perhaps due to the same essential difficulty try
+<<*>>=
+)clear all
+--S 121
+integrate(normalize(sqrt(1+cos(x)),x),x)
+--R
+--R +--------+
+--R x | x 2
+--R 2sin(-) |2cos(-)
+--R 2 \| 2
+--R (1) ------------------
+--R x
+--R cos(-)
+--R 2
+--R Type: Union(Expression
Integer,...)
+--E
+@
+This gets a non-invertible error
--------------------------------------------------------------------
--- bmt/11/17/92 Ifintegrate
--------------------------------------------------------------------
---It seems that one cannot simply parametrize reducible curves.
---Any attempt at parametrization can only follow one branch.
---For curves of the form y**2=a**2*x**2+b*x+c, these are only reducible
---if a**2*x**2+b*x+c is a perfect square. Basically I think we need to
---choose 1 branch in this case. Under some circumstances we could
---try to proceed down both branches and then unify the result, but for
---the time being, perhaps we should just perform the substitution
---y=sqrt(a**2*x**2+b*x+c) whenever that is a perfect square.
---(then we don't even need to do any back subsitution or change to dx).
---
---For another example, perhaps due to the same essential difficulty try
+I have fixed integrate(sqrt(1+cos(x)),x) and
integrate(normalize(sqrt(1+cos(x)),x),x)
--- this gets a non-invertible error
+fix was to function prootintegrate in INTPAF in intaf.spad
---I have fixed integrate(sqrt(1+cos(x)),x) and
---integrate(normalize(sqrt(1+cos(x)),x),x)
---fix was to function prootintegrate in INTPAF in intaf.spad
+james/02/11/93 integrate
+used to give error:
+\begin{verbatim}
+ No identity element for reduce of empty list using operation max
+\end{verbatim}
+<<*>>=
+)clear all
--------------------------------------------------------------------
--- james/02/11/93 integrate
--- used to give error:
--- No identity element for reduce of empty list using operation
--- max
--------------------------------------------------------------------
+--S 122
integrate(((-x-1)*log((x**2+x))**2+2*log(x))/(x+1),x)
+--R
+--R x 2 2
+--I ++ (- %K - 1)log(%K + %K) + 2log(%K)
+--I (1) | ----------------------------------- d%K
+--I ++ %K + 1
+--R Type: Union(Expression
Integer,...)
+--E
--------------------------------------------------------------------
--- dewar/02/15/93
--------------------------------------------------------------------
+@
+dewar/02/15/93
+
+\begin{verbatim}
+ i::Polynomial(Integer)
+ %::Union(Symbol,Polynomial Integer)
+ list %
+\end{verbatim}
+This is a bug in old style union handling, this style of unions
+will become obsolete, so the suggested fix is to use tagged unions instead:
+
+<<*>>=
)clear all
---i::Polynomial(Integer)
---%::Union(Symbol,Polynomial Integer)
---list %
---
---this is a bug in old style union handling, this style of unions
---will become obsolete, so the suggested fix is to use tagged unions instead:
-i::POLY INT
-%::Union(s:Symbol, p:POLY INT)
-list % -- works
--------------------------------------------------------------------
--- grabm/03/04/93
--- parser bug? not for system commands
--------------------------------------------------------------------
+--S 123
+t1:=i::POLY INT
+--R
+--R (1) i
+--R Type: Polynomial
Integer
+--E
+
+--S 124
+t2:=t1::Union(s:Symbol, p:POLY INT)
+--R
+--R (2) i
+--R Type: Union(p: Polynomial
Integer,...)
+--E
+
+--S 125
+list t2
+--R
+--R (3) [i]
+--R Type: List Union(s: Symbol,p: Polynomial
Integer)
+--E
+
+@
+grabm/03/04/93 parser bug? not for system commands
+\begin{verbatim}
)clear all
-- gives error: Sorry, but no option begins with upd;
)load /spad/mnt/rios/algebra/EQ )upd; a:=1
-- works
)load /spad/mnt/rios/algebra/EQ )upd ; a:=1
+\end{verbatim}
--------------------------------------------------------------------
--- bronstein/03/08/93
--------------------------------------------------------------------
+bronstein/03/08/93 wrong particular solution
+<<*>>=
)clear all
+
+--S 126
I:=operator 'I
+--R
+--R (1) I
+--R Type:
BasicOperator
+--E
+
+--S 127
J:=operator 'J
+--R
+--R (2) J
+--R Type:
BasicOperator
+--E
+
+--S 128
eq := mu * D(I x,x) = - (K + S) * I(x) + S*J(x)
+--R
+--R ,
+--R (3) muI (x)= S J(x) + (- S - K)I(x)
+--R
+--R Type: Equation Expression
Integer
+--E
+
+--S 129
solve(eq,I,x)
---> wrong particular solution
-
--------------------------------------------------------------------
--- dewar/03/16/93
--- bug writing backslash to files
---comment:
---This is not a bug, what is written to the file is the lisp representation
---of the string "\\test" (since '\' is lisp escape character).
---The file is read and written using lisp primitives and thus the escape
---characters need to be doubled. This is consistent, since if you read the
---item back in from the file, the correct string is reconstituted.
---
---If you want to create a text file which simply contains lines of text
---instead of lines of lisp strings, use:
---ofile : TextFile := open("test","output") ...
---then you will get a file of literal strings with no string delimiters and
---no escape characters.
--------------------------------------------------------------------
+--R
+--R (4)
+--R (- S - K)x (- S -
K)x
+--R ---------- x
----------
+--I mu ++ S J(%K) mu
+--I [particular= %e | ------------------ d%K ,basis= [%e
]]
+--I ++ - %K S - %K K
+--R -------------
+--R mu
+--R mu %e
+--RType: Union(Record(particular: Expression Integer,basis: List Expression
Integer),...)
+--E
+
+@
+
+dewar/03/16/93 bug writing backslash to files
+
+This is not a bug, what is written to the file is the lisp representation
+of the string "\\test" (since '\' is lisp escape character).
+The file is read and written using lisp primitives and thus the escape
+characters need to be doubled. This is consistent, since if you read the
+item back in from the file, the correct string is reconstituted.
+
+If you want to create a text file which simply contains lines of text
+instead of lines of lisp strings, use:
+\begin{verbatim}
+ ofile : TextFile := open("test","output") ...
+\end{verbatim}
+then you will get a file of literal strings with no string delimiters and
+no escape characters.
+<<*>>=
)clear all
+
+--S 130
ofile: File String := open("/tmp/test","output")
--- this writes "\\\\test" but should write "\\test"
+--R
+--R (1) "/tmp/test"
+--R Type: File
String
+--E
+
+@
+This writes ``\verb|\\\\test|'' but should write ``\verb|\\test|''
+<<*>>=
+
+--S 131
write!(ofile,"\\test"::String)
+--R
+--R (2) "\\test"
+--R Type:
String
+--E
+
+--S 132
close! ofile
+--R
+--R (3) "/tmp/test"
+--R Type: File
String
+--E
+
+@
+
+themos/04/07/93 this sometimes fails
--------------------------------------------------------------------
--- themos/04/07/93
--- this sometimes fails
--- comment:
---thanks for tracing down the problem. Basically the lines for monicizing
---the result need to be done before we try to
---retract back from the algebraic extension.
--------------------------------------------------------------------
+Thanks for tracing down the problem. Basically the lines for monicizing
+the result need to be done before we try to
+retract back from the algebraic extension.
+<<*>>=
)clear all
+
+--S 133
pol:DMP([x,y,z],PF(2)):=x**2*y**2+x**2*y*z+x**2*z**2+x*y*z**2+y**3*z+y*z**3
+--R
+--R 2 2 2 2 2 2 3 3
+--R (1) x y + x y z + x z + x y z + y z + y z
+--R Type: DistributedMultivariatePolynomial([x,y,z],PrimeField
2)
+--E
+
+--S 134
factor pol
+--R
+--R 2 2 2 2 2 2 3 3
+--R (2) x y + x y z + x z + x y z + y z + y z
+--R Type: Factored DistributedMultivariatePolynomial([x,y,z],PrimeField
2)
+--E
+
+@
+williamson/04/21/93
--------------------------------------------------------------------
--- williamson/04/21/93
--- comment:
---I ran into some problems when factoring polynomials over number fields.
---Evidently the interpretor doesn't choose the correct function 'factor'.
---It uses a function from MULTFACT, when it should be using a function from
---SAEFACT.
--------------------------------------------------------------------
+I ran into some problems when factoring polynomials over number fields.
+Evidently the interpretor doesn't choose the correct function 'factor'.
+It uses a function from MULTFACT, when it should be using a function from
+SAEFACT.
+<<*>>=
)clear all
+
+--S 135
up := UP('w,FRAC INT)
+--R
+--R (1) UnivariatePolynomial(w,Fraction Integer)
+--R Type:
Domain
+--E
+
+--S 136
p : up := w**4 + w**3 + w**2 + w + 1
+--R
+--R 4 3 2
+--R (2) w + w + w + w + 1
+--R Type: UnivariatePolynomial(w,Fraction
Integer)
+--E
+
+--S 137
sae := SAE(FRAC INT,up,p)
+--R
+--R (3)
+--R SimpleAlgebraicExtension(Fraction Integer,UnivariatePolynomial(w,Fraction
Int
+--R eger),w**4+w**3+w*w+w+1)
+--R Type:
Domain
+--E
+
+--S 138
q : UP('x,sae) := x**5 - 1
-factor q -- used to report: x**5-1
+--R
+--R 5
+--R (4) x - 1
+--RType: UnivariatePolynomial(x,SimpleAlgebraicExtension(Fraction
Integer,UnivariatePolynomial(w,Fraction Integer),w**4+w**3+w*w+w+1))
+--E
+
+@
+Used to report: x**5-1
+<<*>>=
+--S 139
+factor q
+--R
+--R 2 3 3 2
+--R (5) (x - 1)(x - w)(x - w )(x - w )(x + w + w + w + 1)
+--RType: Factored UnivariatePolynomial(x,SimpleAlgebraicExtension(Fraction
Integer,UnivariatePolynomial(w,Fraction Integer),w**4+w**3+w*w+w+1))
+--E
+
+--S 140
saefact := SAEFACT(up,sae,UP('x,sae))
+--R
+--R (6)
+--R SimpleAlgebraicExtensionAlgFactor(UnivariatePolynomial(w,Fraction
Integer),Si
+--R mpleAlgebraicExtension(Fraction Integer,UnivariatePolynomial(w,Fraction
Integ
+--R
er),w**4+w**3+w*w+w+1),UnivariatePolynomial(x,SimpleAlgebraicExtension(Fracti
+--R on Integer,UnivariatePolynomial(w,Fraction Integer),w**4+w**3+w*w+w+1)))
+--R Type:
Domain
+--E
+
+--S 141
factor(q)$saefact
+--R
+--R 2 3 3 2
+--R (7) (x - 1)(x - w)(x - w )(x - w )(x + w + w + w + 1)
+--RType: Factored UnivariatePolynomial(x,SimpleAlgebraicExtension(Fraction
Integer,UnivariatePolynomial(w,Fraction Integer),w**4+w**3+w*w+w+1))
+--E
+
+@
+
+grabm/04/30/93
--------------------------------------------------------------------
--- grabm/04/30/93
--- comment:
---This bug has been fixed at yorktown, in radix.spad, change line 163, the
---third line of intgroup to be:
--- empty? rest li => intToExpr first(li)
---and then recompile RADIX
--------------------------------------------------------------------
+This bug has been fixed at yorktown, in radix.spad, change line 163, the
+third line of intgroup to be:
+\begin{verbatim}
+ empty? rest li => intToExpr first(li)
+\end{verbatim}
+and then recompile RADIX
+
+Results in 10 and not A
+<<*>>=
)clear all
--- results in 10 and not A
+
+--S 142
10::RadixExpansion(16)
+--R
+--R (1) A
+--R Type: RadixExpansion
16
+--E
+
+@
address@hidden (manuel bronstein)
--------------------------------------------------------------------
--- address@hidden (manuel bronstein)
--------------------------------------------------------------------
+Used to return x, now returns 1
+<<*>>=
)clear all
+
+--S 143
r:=rule 'x == 1
--- used to return x, now returns 1
+--R
+--R (1) x == 1
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 144
r x
+--R
+--R (2) 1
+--R Type: Expression
Integer
+--E
+
+@
address@hidden (tony kennedy)
--------------------------------------------------------------------
--- address@hidden (tony kennedy)
--------------------------------------------------------------------
+<<*>>=
)clear all
-factor(-12)
-%**2
--------------------------------------------------------------------
--- address@hidden (manuel bronstein)
--- >> Error: cannot retract nonconstant polynomial
--------------------------------------------------------------------
+--S 145
+t1:=factor(-12)
+--R
+--R 2
+--R (1) - 2 3
+--R Type: Factored
Integer
+--E
+
+--S 146
+t1**2
+--R
+--R 4 2
+--R (2) 2 3
+--R Type: Factored
Integer
+--E
+
+@
address@hidden (manuel bronstein)
+
+Error: cannot retract nonconstant polynomial
+<<*>>=
)clear all
+
+--S 147
complexNumeric(log(sqrt(-3)))
+--R
+--R (1) 0.5493061443 340548457 + 1.5707963267 948966192 %i
+--R Type: Complex
Float
+--E
+
+@
address@hidden/8/15/93 (Claude Quitte)
+
+Why is it impossible to specify the target/cible types for the
+functions leftProductBy and rightProductBy in the body of the package ???
--------------------------------------------------------------------
--- address@hidden/8/15/93 (Claude Quitte)
--------------------------------------------------------------------
+The code for leftProductBy is correct but the code for rightProductBy
+troubles the compiler !!!! It seems to compile a LOCAL function with
+ANOTHER signature.
+
+\begin{verbatim}
+ initializing nrlib TEST for Test
+ compiling into nrlib TEST
+ compiling exported leftProductBy : Integer -> Integer -> Integer
+Time: 0.35 SEC.
+
+ compiling local rightProductBy : Integer -> Integer
+****** comp fails at level 2 with expression: ******
+error in function rightProductBy
+
+****** level 2 ******
+$x:= #1
+$m:= $EmptyMode
+$f:=
+((((|n| # #) (|rightProductBy| # # #) (|n| #) (|rightProductBy| # # #) ...)))
+
+ >> Apparent user error:
+ no mode found for
+ #1
+ You are being returned to the top level of the interpreter.
+\end{verbatim}
+answer (barry):
+
+Your problem in TEST about specifying target types which are Mapping's
+was a bug in the compiler and has been fixed for Axiom release 2.0.
+
+<<*>>=
+\begin{verbatim}
)clear all
---)abbreviation package TEST Test
---
---Test() : with
---
--- leftProductBy : Integer -> (Integer -> Integer)
--- rightProductBy : Integer -> Mapping(Integer, Integer)
---
--- == add
---
--- leftProductBy(n) == n * #1
---
--- rightProductBy(n : Integer) : Mapping(Integer, Integer) == #1 * n
---------------------------------------------------------------------------
---
---Why is it impossible to specify the target/cible types for the
---functions leftProductBy and rightProductBy in the body of the package ???
---
---The code for leftProductBy is correct but the code for rightProductBy
---troubles the compiler !!!! It seems to compile a LOCAL function with
---ANOTHER signature.
---
---------------------------------------------------------------------------
--- initializing nrlib TEST for Test
--- compiling into nrlib TEST
--- compiling exported leftProductBy : Integer -> Integer -> Integer
---Time: 0.35 SEC.
---
--- compiling local rightProductBy : Integer -> Integer
---****** comp fails at level 2 with expression: ******
---error in function rightProductBy
---
---****** level 2 ******
---$x:= #1
---$m:= $EmptyMode
---$f:=
---((((|n| # #) (|rightProductBy| # # #) (|n| #) (|rightProductBy| # # #) ...)))
---
--- >> Apparent user error:
--- no mode found for
--- #1
--- You are being returned to the top level of the interpreter.
-------------------------------------------------------------------------
--- answer (barry):
---
---Your problem in TEST about specifying target types which are Mapping's
---was a bug in the compiler and has been fixed for Axiom release 2.0.
---
+)sys rm -f /tmp/tpd.spad
+)lisp (setq ofile (open "/tmp/tpd.spad" :direction :output))
+)lisp (format ofile ")abbreviation package TPD Tpd~%")
+)lisp (format ofile "Tpd() : with~%")
+)lisp (format ofile " leftProductBy : Integer -> (Integer -> Integer)~%")
+)lisp (format ofile " rightProductBy : Integer -> Mapping(Integer,
Integer)~%")
+)lisp (format ofile " == add~%")
+)lisp (format ofile " leftProductBy(n) == n * #1~%")
+)lisp (format ofile " rightProductBy(n : Integer) : Mapping(Integer,
Integer) == #1 * n~%")
+)lisp (close ofile)
+)compile /tmp/tpd.spad
+)sys rm -f /tmp/tpd.spad
+)sys rm -rf /tmp/TPD.nrlib
--------------------------------------------------------------------
--- address@hidden/10/6/93 (Manuel Bronstein)
--------------------------------------------------------------------
+@
address@hidden/10/6/93 (Manuel Bronstein)
+
+Here is algfunc.spad with an old bug fixed: eventh-roots of negative floats
+coerced to EXPR FLOAT (e.g. sqrt(-1::EXPR FLOAT)) used to produced an error
+message since the sqrt from FLOAT was called.
+
+I have now fixed iroot from AF(R,)
+to call the sqrt from R only if either R is algebraically closed, or the
+root is odd, or the argument is positive. Here's the new behaviour:
+
+As a side-effect, this fixes the problem with numeric
+(which was instantiating \verb|sqrt(-1)\$EXPR(FLOAT))|.
+<<*>>=
)clear all
---Here is algfunc.spad with an old bug fixed: eventh-roots of negative floats
---coerced to EXPR FLOAT (e.g. sqrt(-1::EXPR FLOAT)) used to produced an error
---message since the sqrt from FLOAT was called.
---I have now fixed iroot from AF(R,)
---to call the sqrt from R only if either R is algebraically closed, or the
---root is odd, or the argument is positive. Here's the new behaviour:
+
+--S 148
sqrt(-1::EXPR FLOAT)
+--R
+--R +-----+
+--R (1) \|- 1.0
+--R Type: Expression
Float
+--E
+
+--S 149
sqrt(2::EXPR FLOAT)
+--R
+--R
+--R (2) 1.4142135623 730950488
+--R Type: Expression
Float
+--E
+
+--S 150
nthRoot(-2::EXPR FLOAT, 3)
+--R
+--R (3) - 1.2599210498 948731648
+--R Type: Expression
Float
+--E
+
+--S 151
nthRoot(-2::EXPR FLOAT, 4)
---As a side-effect, this fixes the problem with numeric
--- (which was instantiating sqrt(-1)$EXPR(FLOAT)).
+--R
+--R 4+-----+
+--R (4) \|- 2.0
+--R Type: Expression
Float
+--E
--------------------------------------------------------------------
--- address@hidden/10/6/93 (Manuel Bronstein)
--------------------------------------------------------------------
+@
address@hidden/10/6/93 (Manuel Bronstein)
+
+Here is an updated efstruc.spad where ker2trigs now knows about abs.
+<<*>>=
)clear all
---Here is an updated efstruc.spad where ker2trigs now knows about abs.
-real abs(4 + %i * 5) -- does the right thing.
--------------------------------------------------------------------
--- address@hidden/10/5/93 (Manuel Bronstein)
--------------------------------------------------------------------
+--S 152
+real abs(4 + %i * 5)
+--R
+--R +--+
+--R (1) \|41
+--R Type: Expression
Integer
+--E
+
+@
address@hidden/10/5/93 (Manuel Bronstein)
+
+Here is again an enhanced elemntry.spad. What happens now is that
+\begin{verbatim}
+ exp(q * %i * %pi)
+\end{verbatim}
+gets automatically simplifed whenever q is a rational number whose
+denominator is one of \{1,2,3,4,6\}.
+<<*>>=
)clear all
---Here is again an enhanced elemntry.spad. What happens now is that
-exp(q * %i * %pi)
---gets automatically simplifed whenever q is a rational number whose
---denominator is one of {1,2,3,4,6}.
+
+--S 153
exp(5/3*%i*%pi)
+--R
+--R +-+
+--R - %i\|3 + 1
+--R (1) ------------
+--R 2
+--R Type: Expression Complex
Integer
+--E
--------------------------------------------------------------------
--- address@hidden/10/4/93 (Manuel Bronstein)
--- address@hidden (Richard Luczak)
--------------------------------------------------------------------
+@
address@hidden/10/4/93 (Manuel Bronstein),\\
address@hidden (Richard Luczak)
+<<*>>=
)clear all
+
+--S 154
exp(log(-1))
+--R
+--R (1) - 1
+--R Type: Expression
Integer
+--E
+
+--S 155
sum((-1)**k * (k+m),k=0..n)
+--R
+--R n
+--R (2n + 2m + 1)(- 1) + 2m - 1
+--R (2) ----------------------------
+--R 4
+--R Type: Expression
Integer
+--E
+
+@
address@hidden/10/4/93 (Manuel Bronstein)
--------------------------------------------------------------------
--- address@hidden/10/4/93 (Manuel Bronstein)
--------------------------------------------------------------------
+Simplifies (abs now checks quotients instead of just retraction to R).
+<<*>>=
)clear all
---simplifies (abs now checks quotients instead of just retraction to R).
+
+--S 156
abs((1/2)::EXPR(INT))
+--R
+--R 1
+--R (1) -
+--R 2
+--R Type: Expression
Integer
+--E
--------------------------------------------------------------------
--- address@hidden/10/4/93 (Manuel Bronstein)
--------------------------------------------------------------------
+@
address@hidden/10/4/93 (Manuel Bronstein)
+<<*>>=
)clear all
+
+--S 157
integrate(1/(x**2 + %i*a),x)
+--R
+--R +--+ +--+ +--+ +--+
+--R |%i |%i |%i |%i
+--R |-- log(%i a |-- + x) - |-- log(- %i a |-- + x)
+--R \| a \| a \| a \| a
+--R (1) ---------------------------------------------------
+--R 2
+--R Type: Union(Expression Complex
Integer,...)
+--E
--------------------------------------------------------------------
--- address@hidden/9/28/93 (Barry Trager)
--------------------------------------------------------------------
+@
address@hidden/9/28/93 (Barry Trager)
+<<*>>=
)clear all
+
+--S 158
limit(1/2**n,n=%plusInfinity)
+--R
+--R (1) 0
+--R Type: Union(OrderedCompletion Expression
Integer,...)
+--E
+
+@
address@hidden/9/22/93 (Manuel Bronstein)
--------------------------------------------------------------------
--- address@hidden/9/22/93 (Manuel Bronstein)
--------------------------------------------------------------------
+Here is an update to efstruc.spad that handles complex constants much better.
+Negative square roots and negative logs are now recognized as complex and
+treated properly by real?, real, imag, and complexForm:
+
+As a result, integrals involving sqrt(-2) etc... are now treated correctly
+(this was the case only for sqrt(-1) with the older version).
+<<*>>=
)clear all
---Here is an update to efstruc.spad that handles complex constants much better.
---Negative square roots and negative logs are now recognized as complex and
---treated properly by real?, real, imag, and complexForm:
+
+--S 159
x := sqrt(-3) + sqrt 2 + sqrt(- exp a) + log(-a**2-1)
+--R
+--R +-----+
+--R | a 2 +-+ +---+
+--R (1) \|- %e + log(- a - 1) + \|2 + \|- 3
+--R Type: Expression
Integer
+--E
+
+--S 160
real? x
+--R
+--R (2) false
+--R Type:
Boolean
+--E
+
+--S 161
real x
+--R
+--R 2 +-+
+--R (3) log(a + 1) + \|2
+--R Type: Expression
Integer
+--E
+
+--S 162
imag x
---As a result, integrals involving sqrt(-2) etc... are now treated correctly
---(this was the case only for sqrt(-1) with the older version).
-
--------------------------------------------------------------------
--- address@hidden/9/22/93 (Manuel Bronstein)
--------------------------------------------------------------------
-)clear all
--->> haha := rule x*x == z
--->> haha(a*a + b*b + c**2) --> 3z
--->> haha(a*a + b*b + c**2 + d*d) --> z
--->>
--->> the bug is that the last line returns z instead of 4z.
---
---Sorry guys, this is not a bug: haha is so general a rule that it matches
---the integer 4 (as 2 squared), so the rewrite chain for the last example is:
---
---a*a + b*b + c**2 + d*d ---> z + z + z + z = 4 * z ---> z * z ---> z
---
---Here is a console showing what exactly happens:
+--R
+--R +---+
+--R | a +-+
+--R (4) \|%e + \|3 + %pi
+--R Type: Expression
Integer
+--E
+
+@
address@hidden/9/22/93 (Manuel Bronstein)
+\begin{verbatim}
+ haha := rule x*x == z
+ haha(a*a + b*b + c**2) --> 3z
+ haha(a*a + b*b + c**2 + d*d) --> z
+\end{verbatim}
+
+The bug is that the last line returns z instead of 4z.
+
+Sorry guys, this is not a bug: haha is so general a rule that it matches
+the integer 4 (as 2 squared), so the rewrite chain for the last example is:
+\begin{verbatim}
+ a*a + b*b + c**2 + d*d ---> z + z + z + z = 4 * z ---> z * z ---> z
+\end{verbatim}
+Here is a console showing what exactly happens:
+<<*>>=
+)clear all
+
+--S 163
haha := rule x*x == z
+--R
+--R 2
+--R (1) x == z
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 164
haha 4
+--R
+--R (2) z
+--R Type: Expression
Integer
+--E
+
+--S 165
haha 3
+--R
+--R (3) 3
+--R Type: Expression
Integer
+--E
+
+--S 166
haha(4*z)
+--R
+--R (4) z
+--R Type: Expression
Integer
+--E
+
+--S 167
haha(3*z)
---To see the whole rewrite chain:
-a*a + b*b + c**2 + d*d
-applyRules([haha], %, 1)$APPRULE(INT,INT,EXPR INT)
-applyRules([haha], %, 1)$APPRULE(INT,INT,EXPR INT)
-applyRules([haha], %, 1)$APPRULE(INT,INT,EXPR INT)
---I think it's actually the proper behavior for the pattern matcher. This
---example shows that rules can bite, something Mma users are quite aware about!
-
--------------------------------------------------------------------
--- dewar/9/16/93 (mike dewar)
--------------------------------------------------------------------
---There's an odd discontinuity about the behaviour of the following
---function:
+--R
+--R (5) 3z
+--R Type: Expression
Integer
+--E
+
+@
+To see the whole rewrite chain:
+<<*>>=
+
+--S 168
+t1:=a*a + b*b + c**2 + d*d
+--R
+--R 2 2 2 2
+--R (6) d + c + b + a
+--R Type: Polynomial
Integer
+--E
+
+--S 169
+t2:=applyRules([haha], t1, 1)$APPRULE(INT,INT,EXPR INT)
+--R
+--R (7) 4z
+--R Type: Expression
Integer
+--E
+
+--S 170
+t3:=applyRules([haha], t2, 1)$APPRULE(INT,INT,EXPR INT)
+--R
+--R 2
+--R (8) z
+--R Type: Expression
Integer
+--E
+
+--S 171
+t4:=applyRules([haha], t3, 1)$APPRULE(INT,INT,EXPR INT)
+--R
+--R (9) z
+--R Type: Expression
Integer
+--E
+
+@
+I think it's actually the proper behavior for the pattern matcher. This
+example shows that rules can bite, something Mma users are quite aware about!
+
+dewar/9/16/93 (mike dewar)
+
+There's an odd discontinuity about the behaviour of the following function:
+
+<<*>>=
+)clear all
+--S 172
harm(1) == 1
+--R Type:
Void
+--E
+
+--S 173
harm(n) == harm(n-1) + 1/n
+--R Type:
Void
+--E
+
+--S 174
harm : Integer -> Fraction Integer
-harm(1023) -- takes a little while, as expected
-harm(1024) -- takes forever?
-
--------------------------------------------------------------------
--- address@hidden/9/1/93 (Clifton Williamson)
--------------------------------------------------------------------
---> we should probably support integrate(%,x) on series types.
---> x could be either the variable of expansion or a parameter variable.
---
---We've discussed this before with regard to the function 'differentiate'.
---The technical problem is that "generalized" power series may have
---"coefficients" involving the series variable:
-series(x**x,x=0)
---In this case, as currently implemented, 'differentiate' returns incorrect
---answers:
-)set mess test off
-differentiate %
-)set mess test on
---The solution we discussed (perhaps "resolution" is a better word) was to
---create a separate series type for generalized series. This would be a
---carbon copy of UPXS, except that differentiate(series,variable(series)) and
---integrate(series,variable(series)) would return an error message. The
---error message could also suggest that the user first apply 'approximate',
---then compute a derivative or integral. We would also have a coercion from
---UPXS to this type. The function 'series' would return a UPXS, when the
---coefficients do not involve the series variable, and a "generalized series",
---when the coefficients involve the series variable.
---
---If this sounds cool to you, I'll go ahead with it when I have time. (I'm
---trying to meet a 16 September deadline for a MEGA '94 submission. I'm still
---getting my results together, so time is tight. Let me know if there is a
---time deadline for the AXIOM code!)
---
---> Also we need better tools for taking finite truncations of series objects.
---> In particular since we now mostly use series(expr,var) to create a series,
---> UPXS if over a coefficient domain which supports the symbol of expansion,
---> should be able to truncate a series to an EXPR.
---
---I thought I had you on this one! The signature is there (in pscat.spad):
---
--- if Coef has coerce: Symbol -> Coef then
--- if Coef has "**":(Coef,Expon) -> Coef then
--- approximate: ($,Expon) -> Coef
--- ++ \spad{approximate(f)} returns a truncated power series with the
--- ++ series variable viewed as an element of the coefficient domain.
---
---It certainly works for Laurent series:
-laurent(cos(a+x)/x,x=0)
-approximate(%,3)
---But, unfortunately, I never implemented it for Puiseux series:
-puiseux(cos(a+x)/x,x=0)
-approximate(%,3)
-series(cos(x**(2/3) + a),x=0)
-approximate(%,2)
-approximate(%% 1,7/5)
-
-
--------------------------------------------------------------------
--- address@hidden/10/6/93 (manuel bronstein)
--------------------------------------------------------------------
-)clear all
-)set message type off
-
--- Here is algfunc.spad with an old bug fixed: eventh-roots of negative floats
--- coerced to EXPR FLOAT (e.g. sqrt(-1::EXPR FLOAT)) used to produced an error
--- message since the sqrt from FLOAT was called.
--- I have now fixed iroot from AF(R,)
--- to call the sqrt from R only if either R is algebraically closed, or the
--- root is odd, or the argument is positive. Here's the new behaviour:
+--R Type:
Void
+--E
+
+--S 175
+harm(1023)
+--R Compiling function harm with type Integer -> Fraction Integer
+--R Compiling function harm as a recurrence relation.
+--R
+--R (4)
+--R
56918929836783747778186562994910417364284519936484803738256503363047591584_
+--R
2733402533503384016070102378254838554019021072460441324115730731899754524_
+--R
3051637047716166927952839924926898113855910441901034429495456850574721819_
+--R
0959291999714039498339199810512595613399980225816207570338522992407348206_
+--R
8458996501642275777927445608416707051265982507444007863105751583310646982_
+--R
3197880912123301992627534179914738912810939235218743092316208657821544413_
+--R 7507111
+--R /
+--R
75809030906834453523777687598142141106414192506285711254121803198491495253_
+--R
8828569124484088941284219071465702944545255084351101237230340026626588247_
+--R
6367291194600696876416243191521584529737974727209031624214028323361847170_
+--R
9764957043186933388156027004157275042770718773974970630837005685761111898_
+--R
3824612373575784564285126841732340269099782510311467722310664215128464192_
+--R
0090417465131869897386008931792841181410706392968052396422445537614139284_
+--R 160000
+--R Type: Fraction
Integer
+--E
+
+--S 176
+harm(1024)
+--R
+--R (5)
+--R
11385266612491648666057698881942984686550388639494349140449232739078238853_
+--R
8713454502191389383076154933054707275314439864005454497269185801458595955_
+--R
4127443958199377930451435489735214028719028533199097687298001800355635024_
+--R
8967439746760057644460577384504944069458175160764295712712838133748769663_
+--R
0859452124788876557627760315566968995024077356615956093475101696822581124_
+--R
2802077948390776255359440156469896861491512274902986275736609357453715408_
+--R 90803597
+--R /
+--R
15161806181366890704755537519628428221282838501257142250824360639698299050_
+--R
7765713824896817788256843814293140588909051016870220247446068005325317649_
+--R
5273458238920139375283248638304316905947594945441806324842805664672369434_
+--R
1952991408637386677631205400831455008554143754794994126167401137152222379_
+--R
6764922474715156912857025368346468053819956502062293544462132843025692838_
+--R
4018083493026373979477201786358568236282141278593610479284489107522827856_
+--R 8320000
+--R Type: Fraction
Integer
+--E
+
+@
address@hidden/9/1/93 (Clifton Williamson)
+
+We should probably support integrate(%,x) on series types.
+x could be either the variable of expansion or a parameter variable.
+
+We've discussed this before with regard to the function 'differentiate'.
+The technical problem is that ``generalized'' power series may have
+``coefficients'' involving the series variable:
+<<*>>=
+)clear all
+
+--S 177
+t1:=series(x**x,x=0)
+--R
+--R (1)
+--R 2 3 4 5
+--R log(x) 2 log(x) 3 log(x) 4 log(x) 5
+--R 1 + log(x)x + ------- x + ------- x + ------- x + ------- x
+--R 2 6 24 120
+--R +
+--R 6 7 8 9 10
+--R log(x) 6 log(x) 7 log(x) 8 log(x) 9 log(x) 10
11
+--R ------- x + ------- x + ------- x + ------- x + -------- x + O(x
)
+--R 720 5040 40320 362880 3628800
+--R Type: GeneralUnivariatePowerSeries(Expression
Integer,x,0)
+--E
+
+@
+In this case, as currently implemented, 'differentiate' returns incorrect
+answers:
+<<*>>=
+
+--S 178
+differentiate t1
+
+ >> Error detected within library code:
+ "'differentiate' unavailable on this domain; use 'approximate' first"
+
+--E
+
+@
+The solution we discussed (perhaps ``resolution'' is a better word) was to
+create a separate series type for generalized series. This would be a
+carbon copy of UPXS, except that differentiate(series,variable(series)) and
+integrate(series,variable(series)) would return an error message. The
+error message could also suggest that the user first apply 'approximate',
+then compute a derivative or integral. We would also have a coercion from
+UPXS to this type. The function 'series' would return a UPXS, when the
+coefficients do not involve the series variable, and a ``generalized series'',
+when the coefficients involve the series variable.
+
+If this sounds cool to you, I'll go ahead with it when I have time. (I'm
+trying to meet a 16 September deadline for a MEGA '94 submission. I'm still
+getting my results together, so time is tight. Let me know if there is a
+time deadline for the AXIOM code!)
+
+Also we need better tools for taking finite truncations of series objects.
+In particular since we now mostly use series(expr,var) to create a series,
+UPXS if over a coefficient domain which supports the symbol of expansion,
+should be able to truncate a series to an EXPR.
+
+I thought I had you on this one! The signature is there (in pscat.spad):
+\begin{verbatim}
+ if Coef has coerce: Symbol -> Coef then
+ if Coef has "**":(Coef,Expon) -> Coef then
+ approximate: ($,Expon) -> Coef
+ ++ \spad{approximate(f)} returns a truncated power series with the
+ ++ series variable viewed as an element of the coefficient domain.
+\end{verbatim}
+It certainly works for Laurent series:
+<<*>>=
+)clear all
+
+--S 179
+t1:=laurent(cos(a+x)/x,x=0)
+--R
+--R (1)
+--R - 1 cos(a) sin(a) 2 cos(a) 3 sin(a) 4
+--R cos(a)x - sin(a) - ------ x + ------ x + ------ x - ------ x
+--R 2 6 24 120
+--R +
+--R cos(a) 5 sin(a) 6 cos(a) 7 sin(a) 8 cos(a) 9 10
+--R - ------ x + ------ x + ------ x - ------ x - ------- x + O(x )
+--R 720 5040 40320 362880 3628800
+--R Type: UnivariateLaurentSeries(Expression
Integer,x,0)
+--E
+
+--S 180
+approximate(t1,3)
+--R
+--R 3 4 2
+--R (4x - 24x)sin(a) + (x - 12x + 24)cos(a)
+--R (2) ------------------------------------------
+--R 24x
+--R Type: Expression
Integer
+--E
+
+@
+But, unfortunately, I never implemented it for Puiseux series:
+<<*>>=
+
+--S 181
+t2:=puiseux(cos(a+x)/x,x=0)
+--R
+--R (3)
+--R - 1 cos(a) sin(a) 2 cos(a) 3 sin(a) 4
+--R cos(a)x - sin(a) - ------ x + ------ x + ------ x - ------ x
+--R 2 6 24 120
+--R +
+--R cos(a) 5 sin(a) 6 cos(a) 7 sin(a) 8 cos(a) 9 10
+--R - ------ x + ------ x + ------ x - ------ x - ------- x + O(x )
+--R 720 5040 40320 362880 3628800
+--R Type: UnivariatePuiseuxSeries(Expression
Integer,x,0)
+--E
+
+--S 182
+approximate(t2,3)
+--R
+--R 3 4 2
+--R (4x - 24x)sin(a) + (x - 12x + 24)cos(a)
+--R (4) ------------------------------------------
+--R 24x
+--R Type: Expression
Integer
+--E
+
+--S 183
+t3:=series(cos(x**(2/3) + a),x=0)
+--R
+--R (5)
+--R 2 4 8 10
11
+--R - - - --
--
+--R 3 cos(a) 3 sin(a) 2 cos(a) 3 sin(a) 3
3
+--R cos(a) - sin(a)x - ------ x + ------ x + ------ x - ------ x + O(x
)
+--R 2 6 24 120
+--R Type: UnivariatePuiseuxSeries(Expression
Integer,x,0)
+--E
+
+--S 184
+approximate(t3,2)
+--R
+--R 3+-+2 3+-+ 2
+--R - 6sin(a)\|x - 3x cos(a)\|x + x sin(a) + 6cos(a)
+--R (6) ---------------------------------------------------
+--R 6
+--R Type: Expression
Integer
+--E
+
+--S 185
+approximate(t1,7)
+--R
+--R (7)
+--R 7 5 3
+--R (8x - 336x + 6720x - 40320x)sin(a)
+--R +
+--R 8 6 4 2
+--R (x - 56x + 1680x - 20160x + 40320)cos(a)
+--R /
+--R 40320x
+--R Type: Expression
Integer
+--E
+
+@
address@hidden/10/6/93 (manuel bronstein)
+
+Here is algfunc.spad with an old bug fixed: eventh-roots of negative floats
+coerced to EXPR FLOAT (e.g. sqrt(-1::EXPR FLOAT)) used to produced an error
+message since the sqrt from FLOAT was called.
+
+I have now fixed iroot from AF(R,)
+to call the sqrt from R only if either R is algebraically closed, or the
+root is odd, or the argument is positive. Here's the new behaviour:
+<<*>>=
+)clear all
+
+--S 186
sqrt(-1::EXPR FLOAT)
+--R
+--R +-----+
+--R (1) \|- 1.0
+--R Type: Expression
Float
+--E
+
+--S 187
sqrt(2::EXPR FLOAT)
+--R
+--R (2) 1.4142135623 730950488
+--R Type: Expression
Float
+--E
+
+--S 188
nthRoot(-2::EXPR FLOAT, 3)
+--R
+--R (3) - 1.2599210498 948731648
+--R Type: Expression
Float
+--E
+
+--S 189
nthRoot(-2::EXPR FLOAT, 4)
---As a side-effect, this fixes the problem with numeric
--- (which was instantiating sqrt(-1)$EXPR(FLOAT)).
+--R
+--R 4+-----+
+--R (4) \|- 2.0
+--R Type: Expression
Float
+--E
--------------------------------------------------------------------
--- address@hidden/9/22/93 (manuel bronstein)
--------------------------------------------------------------------
-)clear all
---Here is a (rather major) bug fix to ffactor in FSUPFACT.nrlib. It causes
---a large family of integrals to return 0, because ffactor(?**2+expr) returned
---?**2 when expr involved a parameter. This is fixed now.
+@
+As a side-effect, this fixes the problem with numeric
+(which was instantiating sqrt(-1)\$EXPR(FLOAT)).
+
address@hidden/9/22/93 (manuel bronstein)
+
+Here is a (rather major) bug fix to ffactor in FSUPFACT.nrlib. It causes
+a large family of integrals to return 0, because ffactor(?**2+expr) returned
+?**2 when expr involved a parameter. This is fixed now.
+
address@hidden/8/15/93 James Davenport
+
+\begin{verbatim}
+ (1)
+ VCONCAT
+ log(x)
+ atan(---------)
+ +------+
+ | 2
+ 2 \|a - 1
+ ((- a + 1<0) -> ---------------)
+ +------+
+ | 2
+ \|a - 1
+ ,
+ PAREN
+ 2
+ (- a + 1>0)
+ ->
+ +--------+
+ 2 2 | 2 2
+ log((log(x) - a + 1)\|- a + 1 + (2a - 2)log(x))
+ +
+ 2 2
+ - log(log(x) + a - 1)
+ /
+ +--------+
+ | 2
+ 2\|- a + 1
+ ,
+ 1
+ ( Or (a - 1=0) -> - ------)
+ (a + 1=0) log(x)
+ Type: ConditionalExpression Integer
+
+\end{verbatim}
+is still fairly ugly.
--------------------------------------------------------------------
--- address@hidden/8/15/93 James Davenport
--------------------------------------------------------------------
+<<*>>=
)clear all
+
+--S 190
integrate(1/(x*(log(x)**2+a**2-1)),x)
--- (1)
--- VCONCAT
--- log(x)
--- atan(---------)
--- +------+
--- | 2
--- 2 \|a - 1
--- ((- a + 1<0) -> ---------------)
--- +------+
--- | 2
--- \|a - 1
--- ,
--- PAREN
--- 2
--- (- a + 1>0)
--- ->
--- +--------+
--- 2 2 | 2 2
--- log((log(x) - a + 1)\|- a + 1 + (2a - 2)log(x))
--- +
--- 2 2
--- - log(log(x) + a - 1)
--- /
--- +--------+
--- | 2
--- 2\|- a + 1
--- ,
--- 1
--- ( Or (a - 1=0) -> - ------)
--- (a + 1=0) log(x)
--- Type: ConditionalExpression Integer
---
---is still fairly ugly.
+--R
+--R (1)
+--R +--------+
+------+
+--R 2 2 | 2 2 | 2
+--R (log(x) - a + 1)\|- a + 1 + (2a - 2)log(x) log(x)\|a - 1
+--R log(-----------------------------------------------)
atan(---------------)
+--R 2 2 2
+--R log(x) + a - 1 a - 1
+--R
[----------------------------------------------------,---------------------]
+--R +--------+ +------+
+--R | 2 | 2
+--R 2\|- a + 1 \|a - 1
+--R Type: Union(List Expression
Integer,...)
+--E
--------------------------------------------------------------------
--- address@hidden/8/9/93 (manuel bronstein)
--------------------------------------------------------------------
+@
address@hidden/8/9/93 (manuel bronstein)
+
+Here is efstruc.spad with a change to normalize so that
+this now returns 2**(1/4) ** 2 + 2**(1/4)
+<<*>>=
)clear all
---Here is efstruc.spad with a change to normalize so that
-normalize(2**(1/2) + 2**(1/4)) now returns 2**(1/4) ** 2 + 2**(1/4).
+--S 191
+normalize(2**(1/2) + 2**(1/4))
+--R
+--R 4+-+2 4+-+
+--R (1) \|2 + \|2
+--R Type: Expression
Integer
+--E
--------------------------------------------------------------------
--- address@hidden/8/7/93 satoshi hamaguchi
--------------------------------------------------------------------
+@
+
address@hidden/8/7/93 satoshi hamaguchi
+
+In axiom, both of these yield a warning ``potentialPole''.
+If I use ``noPole'' they give the right answer.
+Isn't this a bug?
+<<*>>=
)clear all
--- In axiom, both
- integrate(%e**x,x=0..1)
---and
- integrate(log(x),x=1..2)
---yield a warning potentialPole .
---If I use "noPole", they give right answers. Isn't this a bug?
---... Satoshi
--------------------------------------------------------------------
--- address@hidden/8/5/93 (manuel bronstein)
--------------------------------------------------------------------
+--S 912
+integrate(%e**x,x=0..1)
+--R
+--R (1) %e - 1
+--R Type: Union(f1: OrderedCompletion Expression
Integer,...)
+--E
+
+--S 913
+integrate(log(x),x=1..2)
+--R
+--R (2) log(4) - 1
+--R Type: Union(f1: OrderedCompletion Expression
Integer,...)
+--E
+
+@
+
address@hidden/8/5/93 (manuel bronstein)
+
+This will return 2**(5/6).
+<<*>>=
)clear all
-simplify(2**(1/3)*2**(1/2)) -- will return 2**(5/6).
--------------------------------------------------------------------
--- address@hidden/8/4/93 (manuel bronstein)
--------------------------------------------------------------------
+--S 194
+simplify(2**(1/3)*2**(1/2)) --
+--R
+--R 6+-+5
+--R (1) \|2
+--R Type: Expression
Integer
+--E
+
+@
address@hidden/8/4/93 (manuel bronstein)
+<<*>>=
)clear all
+
+--S 195
integrate(1/sqrt(1+cos(x)), x)
+--R
+--R +-+ +----------+ 2
+--R +-+ 2\|2 sin(x)\|cos(x) + 1 - cos(x) + 2cos(x) + 3
+--R \|2 log(------------------------------------------------)
+--R 2
+--R cos(x) + 2cos(x) + 1
+--R (1) ---------------------------------------------------------
+--R 2
+--R Type: Union(Expression
Integer,...)
+--E
--------------------------------------------------------------------
--- address@hidden/8/4/93 (manuel bronstein)
--------------------------------------------------------------------
+@
address@hidden/8/4/93 (manuel bronstein)
+
+This used to be division by 0 error
+<<*>>=
)clear all
-normalize atan(cos(x)/sin(x)) -- used to be division by 0 error
--------------------------------------------------------------------
--- address@hidden/8/4/93 (manuel bronstein)
--------------------------------------------------------------------
+--S 196
+normalize atan(cos(x)/sin(x))
+--R
+--R - 2x + %pi
+--R (1) ----------
+--R 2
+--R Type: Expression
Integer
+--E
+
+@
+
address@hidden/8/4/93 (manuel bronstein)
+<<*>>=
)clear all
+
+--S 197
a := 2**(1/6)
+--R
+--R 6+-+
+--R (1) \|2
+--R Type:
AlgebraicNumber
+--E
+
+--S 198
[a**n for n in 2..13]
+--R
+--R 6+-+2 6+-+3 6+-+4 6+-+5 6+-+ 6+-+2 6+-+3 6+-+4 6+-+5 6+-+
+--R (2) [\|2 ,\|2 ,\|2 ,\|2 ,2,2\|2 ,2\|2 ,2\|2 ,2\|2 ,2\|2 ,4,4\|2
]
+--R Type: List
AlgebraicNumber
+--E
+
+@
--------------------------------------------------------------------
--- address@hidden/8/4/93 (manuel bronstein)
--------------------------------------------------------------------
address@hidden/8/4/93 (manuel bronstein)
+<<*>>=
)clear all
+
+--S 199
int:=sqrt(a*(1-u**2)/(1+u**2))/u
-integrate(eval(int,a=1),u) -- works
-integrate(eval(int,a=sqrt(-1)),u) -- seems to run forever
-integrate(eval(int,a=1)*(-1)**(1/4),u) -- dies after a long time
- -- with an elt index error.
+--R
+--R +----------+
+--R | 2
+--R |- a u + a
+--R |----------
+--R | 2
+--R \| u + 1
+--R (1) -------------
+--R u
+--R Type: Expression
Integer
+--E
+
+--S 200
+integrate(eval(int,a=1),u)
+--R
+--R +--------+ +--------+
+--R | 2 | 2
+--R 2 |- u + 1 2 |- u + 1
+--R (u + 1) |-------- - 1 (u + 1) |-------- - 1
+--R | 2 | 2
+--R \| u + 1 \| u + 1
+--R log(-----------------------) + 2atan(-----------------------)
+--R 2 2
+--R u u
+--R (2) -------------------------------------------------------------
+--R 2
+--R Type: Union(Expression
Integer,...)
+--E
+
+@
+Runs forever?
+<<*>>=
+--S 201
+integrate(eval(int,a=sqrt(-1)),u)
+--R
+--R (3)
+--R +---+ +-+
+--R (- \|- 1 + 1)\|2
+--R *
+--R +----------------+
+--R | 2 +---+
+--R 2 +---+ +-+ |(- u + 1)\|- 1 2 +---+ 2
+--R (u + 1)\|- 1 \|2 |---------------- + (u - 1)\|- 1 + u + 1
+--R | 2
+--R \| u + 1
+--R
log(---------------------------------------------------------------)
+--R 2 +---+ +-+
+--R u \|- 1 \|2
+--R +
+--R +----------------+
+--R | 2 +---+
+--R 2 +---+ +-+ |(- u + 1)\|- 1 +---+
+--R (u + 1)\|- 1 \|2 |---------------- - \|- 1 +
1
+--R | 2
+--R +---+ +-+ \| u + 1
+--R (\|- 1 + 1)\|2
log(--------------------------------------------------)
+--R 2 +---+ +-+
+--R u \|- 1 \|2
+--R +
+--R +---+ +-+
+--R (\|- 1 - 1)\|2
+--R *
+--R +----------------+
+--R | 2 +---+
+--R 2 +---+ +-+ |(- u + 1)\|- 1 2 +---+ 2
+--R (u + 1)\|- 1 \|2 |---------------- + (- u - 1)\|- 1 - u
+ 1
+--R | 2
+--R \| u + 1
+--R
log(-----------------------------------------------------------------)
+--R 2 +---+ +-+
+--R u \|- 1 \|2
+--R /
+--R 4
+--R Type: Union(Expression
Integer,...)
+--E
--------------------------------------------------------------------
--- address@hidden/8/4/93 (manuel bronstein)
--------------------------------------------------------------------
+@
+Dies after a long time with an elt index error
+<<*>>=
+--S 202
+integrate(eval(int,a=1)*(-1)**(1/4),u)
+--R
+--R (4)
+--R +--------+
+--R | 2
+--R 2 +---+ |- u + 1 +---+ 2
+--R (u + 1)\|- 1 |-------- - \|- 1 + u
+--R | 2
+--R +---+ +-+ \| u + 1
+--R (- \|- 1 + 1)\|2 log(---------------------------------------)
+--R 2 +---+
+--R u \|- 1
+--R +
+--R +--------+
+--R | 2
+--R 2 |- u + 1
+--R (u + 1) |-------- - 1
+--R | 2
+--R +---+ +-+ \| u + 1
+--R (\|- 1 + 1)\|2 log(-----------------------)
+--R 2
+--R u
+--R +
+--R +--------+
+--R | 2
+--R 2 +---+ |- u + 1 +---+ 2
+--R (u + 1)\|- 1 |-------- - \|- 1 - u
+--R | 2
+--R +---+ +-+ \| u + 1
+--R (\|- 1 - 1)\|2 log(---------------------------------------)
+--R 2 +---+
+--R u \|- 1
+--R /
+--R 4
+--R Type: Union(Expression
Integer,...)
+--E
+
+@
address@hidden/8/4/93 (manuel bronstein)
+<<*>>=
)clear all
-sqrt((1-x**2)*(1-k**2*x**2))
-integrate(x/%,x)
--------------------------------------------------------------------
--- address@hidden/7/26/93 (manuel bronstein)
--------------------------------------------------------------------
+--S 203
+t1:=sqrt((1-x**2)*(1-k**2*x**2))
+--R
+--R +-----------------------+
+--R | 2 4 2 2
+--R (1) \|k x + (- k - 1)x + 1
+--R Type: Expression
Integer
+--E
+
+--S 204
+integrate(x/t1,x)
+--R
+--R +-----------------------+
+--R | 2 4 2 2 2 2 2
+--R log(2k\|k x + (- k - 1)x + 1 - 2k x + k + 1)
+--R (2) - --------------------------------------------------
+--R 2k
+--R Type: Union(Expression
Integer,...)
+--E
+
+@
address@hidden/7/26/93 (manuel bronstein)
+<<*>>=
)clear all
-rad:=last zerosOf((2+y)**8-3,y)
-k:=first kernels %
-eval(rad,k,rad)
--------------------------------------------------------------------
--- address@hidden/7/15/93 themos tsikas
--------------------------------------------------------------------
+--S 205
+t1:=last zerosOf((2+y)**8-3,y)
+--R
+--R +-------------+
+--R | +-------+
+--R | | +-+
+--R - \|- 2\|- 4\|3 - 4
+--R (1) ----------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 206
+k:=first kernels t1
+--R
+--R +-------------+
+--R | +-------+
+--R | | +-+
+--R (2) \|- 2\|- 4\|3
+--R Type: Kernel Expression
Integer
+--E
+
+--S 207
+eval(t1,k,t1)
+--R
+--R +-------------+
+--R | +-------+
+--R | | +-+
+--R \|- 2\|- 4\|3 - 4
+--R (3) --------------------
+--R 4
+--R Type: Expression
Integer
+--E
+
+@
address@hidden/7/15/93 themos tsikas
+<<*>>=
)clear all
+
+--S 208
f := (x - y) / (x + y)
+--R
+--R - y + x
+--R (1) -------
+--R y + x
+--R Type: Fraction Polynomial
Integer
+--E
+
+--S 209
eval(f,x=1/x)
+--R
+--R - x y + 1
+--R (2) ---------
+--R x y + 1
+--R Type: Fraction Polynomial
Integer
+--E
+
+@
address@hidden/7/15/93 themos tsikas
--------------------------------------------------------------------
--- address@hidden/7/15/93 themos tsikas
--------------------------------------------------------------------
+I claim that these two should give the same result (they do in Reduce):
+(precision was 200)
+
+<<*>>=
)clear all
+
+--S 210
digits 200
---I claim that these two should give the same result (they do in Reduce):
---(precision was 200)
+--R
+--R (1) 20
+--R Type:
PositiveInteger
+--E
+
+--S 211
a:=4*sin(2*%pi/9)*sin(5*%pi/9)/sqrt(3)
+--R
+--R 2%pi 5%pi
+--R 4sin(----)sin(----)
+--R 9 9
+--R (2) -------------------
+--R +-+
+--R \|3
+--R Type: Expression
Integer
+--E
+
+--S 212
b:=1/(2*sin(%pi/9))
+--R
+--R 1
+--R (3) ---------
+--R %pi
+--R 2sin(---)
+--R 9
+--R Type: Expression
Integer
+--E
+
+--S 213
a::EXPR FLOAT
-b::EXPR FLOAT
-
--------------------------------------------------------------------
--- address@hidden/9/15/93 (Barry Trager)
--------------------------------------------------------------------
+--R
+--R (4)
+--R 1.4619022000 8154362611 6377206683 1458519367 5283075946 2240855318
493177672
+--R 5 8139967590 4919627790 5155131563 5927196029 5136978338 4351923544
366777748
+--R 4 3145835284 9347399264 0517928099 4530246377 7454860377 762537213
+--R Type: Expression
Float
+--E
+
+--S 214
+b::EXPR FLOAT
+--R
+--R (5)
+--R 1.4619022000 8154362611 6377206683 1458519367 5283075946 2240855318
493177672
+--R 5 8139967590 4919627790 5155131563 5927196029 5136978338 4351923544
366777748
+--R 4 3145835284 9347399264 0517928099 4530246377 7454860377 762537213
+--R Type: Expression
Float
+--E
+
+--S 215
+digits 20
+--R
+--R (6) 200
+--R Type:
PositiveInteger
+--E
+@
address@hidden/9/15/93 (Barry Trager)
+<<*>>=
)clear all
+
+--S 216
limit(tanh(x),x=%plusInfinity)
+--R
+--R (1) 1
+--R Type: Union(OrderedCompletion Expression
Integer,...)
+--E
--------------------------------------------------------------------
--- address@hidden/6/14/93 (manuel bronstein)
--------------------------------------------------------------------
+@
address@hidden/6/14/93 (manuel bronstein)
+<<*>>=
)clear all
+
+--S 217
a := x::EXPR INT
+--R
+--R (1) x
+--R Type: Expression
Integer
+--E
+
+--S 218
b := x::EXPR COMPLEX INT
+--R
+--R (2) x
+--R Type: Expression Complex
Integer
+--E
+
+--S 219
zeroOf(a**4+1,x)
+--R
+--R +---+
+--R \|- 1 + 1
+--R (3) ----------
+--R +-+
+--R \|2
+--R Type: Expression
Integer
+--E
+
+--S 220
zeroOf(b**4+1,x)
+--R
+--R 1 + %i
+--R (4) ------
+--R +-+
+--R \|2
+--R Type: Expression Complex
Integer
+--E
+
+@
address@hidden/6/11/93 (manuel bronstein)
+
+Now returns 0 (was crashing before)
+<<*>>=
+)clear all
+
+--S 221
+normalize(0**a)
+--R
+--R (1) 0
+--R Type: Expression
Integer
+--E
--------------------------------------------------------------------
--- address@hidden/6/11/93 (manuel bronstein)
--------------------------------------------------------------------
+@
address@hidden/6/11/93 (manuel bronstein)
+
+New export ``complexForm(f)'' which returns the rectangular complex form of f:
+<<*>>=
)clear all
-normalize(0**a) -- now returns 0 (was crashing before)
--------------------------------------------------------------------
--- address@hidden/6/11/93 (manuel bronstein)
--------------------------------------------------------------------
+--S 222
+t1:=log(a+%i * b)
+--R
+--R (1) log(%i b + a)
+--R Type: Expression Complex
Integer
+--E
+
+--S 223
+complexForm t1
+--R
+--R 2 2
+--R log(b + a ) b
+--R (2) ------------ + atan(-)%i
+--R 2 a
+--R Type: Complex Expression
Integer
+--E
+
+--S 224
+t2:=complexIntegrate(1/(x-%i),x)
+--R
+--R (3) log(x - %i)
+--R Type: Expression Complex
Integer
+--E
+
+--S 225
+complexForm t2
+--R
+--R 2
+--R log(x + 1) 1
+--R (4) ----------- - atan(-)%i
+--R 2 x
+--R Type: Complex Expression
Integer
+--E
+
+@
address@hidden
+
+normalForm est faux pour les polynomes a coef dans un anneau:
+<<*>>=
)clear all
-- new export complexForm(f) which returns the rectangular complex form of f:
-log(a+%i * b)
-complexForm %
-complexIntegrate(1/(x-%i),x)
-complexForm %
--------------------------------------------------------------------
--- address@hidden
--------------------------------------------------------------------
--- normalForm est faux pour les polynomes a coef dans un anneau:
+--S 226
D:=DMP([a,b],INT)
+--R
+--R (1) DistributedMultivariatePolynomial([a,b],Integer)
+--R Type:
Domain
+--E
+
+--S 227
l:List D:=[2*a-1,b-a]
+--R
+--R (2) [2a - 1,- a + b]
+--R Type: List
DistributedMultivariatePolynomial([a,b],Integer)
+--E
+
+--S 228
g:=groebner l
+--R
+--R (3) [2a - 1,2b - 1]
+--R Type: List
DistributedMultivariatePolynomial([a,b],Integer)
+--E
+
+--S 229
p:D:=a
+--R
+--R (4) a
+--R Type:
DistributedMultivariatePolynomial([a,b],Integer)
+--E
+
+@
+Was 1, au lieu de 1/2 !!
+<<*>>=
+--S 230
normalForm(p,g)
--- was 1
--- au lieu de 1/2 !!
+--R
+--R 1
+--R (5) -
+--R 2
+--R Type: DistributedMultivariatePolynomial([a,b],Fraction
Integer)
+--E
--------------------------------------------------------------------
--- address@hidden/6/9/93 themos tsikas
--------------------------------------------------------------------
+@
address@hidden/6/9/93 themos tsikas
+<<*>>=
)clear all
+
+--S 231
squareFreePart(50)
+--R
+--R (1) 10
+--R Type:
PositiveInteger
+--E
+
+@
--------------------------------------------------------------------
--- address@hidden/6/4/92 mike dewar
--------------------------------------------------------------------
address@hidden/6/4/92 mike dewar
+<<*>>=
)clear all
+
+--S 232
f:=complexIntegrate(1/((x-%i)*(x-2*%i)),x)
+--R
+--R (1) %i log(x - %i) - %i log(x - 2%i)
+--R Type: Expression Complex
Integer
+--E
+
+--S 233
g:=subst(f,x=1)
+--R
+--R (2) %i log(1 - %i) - %i log(1 - 2%i)
+--R Type: Expression Complex
Integer
+--E
+
+--S 234
imag g
+--R
+--R - log(5) + log(2)
+--R (3) -----------------
+--R 2
+--R Type: Expression
Integer
+--E
--------------------------------------------------------------------
--- daly
--------------------------------------------------------------------
+@
+daly - not wrong but the order is obscure...
+<<*>>=
)clear all
--- not wrong but the order is obscure...
+
+--S 235
primes(1,10)
+--R
+--R (1) [2,7,5,3]
+--R Type: List
Integer
+--E
--------------------------------------------------------------------
--- address@hidden/1/11/93 themos tsikas
--------------------------------------------------------------------
+@
address@hidden/1/11/93 themos tsikas
+<<*>>=
)clear all
+
+--S 236
m:=matrix[[1,a,0],[b,0,a],[0,b,c]]
+--R
+--R +1 a 0+
+--R | |
+--R (1) |b 0 a|
+--R | |
+--R +0 b c+
+--R Type: Matrix Polynomial
Integer
+--E
+
+@
+This ends in error:
+Internal system problem in function appInfix : fewer than 2 arguments to
+an infix function
+<<*>>=
+--S 237
ll:=radicalEigenvectors m
--- ends in error:
--- Internal system problem in function appInfix : fewer than 2 arguments to
--- an infix function
---
--- but this works
+--R
+--R (2)
+--R [
+--R [
+--R radval =
+--R 9
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c
+--R +
+--R 3 3 2 2
+--R - 32a b - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R +
+--R (3c + 3)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c -
32a b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R +
+--R 2
+--R c - c + 6a b + 1
+--R /
+--R 9
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R ,
+--R radmult= 1,
+--R
+--R radvect =
+--R [
+--R [
+--R [
+--R +-+
+--R (- 18c + 18a b + 18)\|3
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c -
32a b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R +
+--R 9
+--R *
+--R ROOT
+--R 4 3 2 2
2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b - 1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 2 +-+
+--R (3c + (- 21a b - 9)c + 15a b + 6)\|3
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R +
+--R (- 3c + 6)
+--R *
+--R ROOT
+--R 4 3 2 2 2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b - 1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 3 2 2 2 +-+
+--R (c + (11a b + 1)c + (- 11a b - 4)c + 24a b + 2a b + 2)\|3
+--R /
+--R 2 +-+
+--R 54b \|3
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R ]
+--R ,
+--R
+--R [
+--R 9
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c -
32a b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R +
+--R (- 6c + 3)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R +
+--R 2
+--R c - c + 6a b + 1
+--R /
+--R 9b
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3 2 2
2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b -
1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R ]
+--R ,
+--R [1]]
+--R ]
+--R ]
+--R ,
+--R
+--R [
+--R radval =
+--R +---+
+--R (- 9\|- 3 - 9)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c
+--R +
+--R 3 3 2 2
+--R - 32a b - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R +
+--R +---+
+--R ((3c + 3)\|- 3 - 3c - 3)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c -
32a b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R +
+--R 2
+--R 2c - 2c + 12a b + 2
+--R /
+--R +---+
+--R (9\|- 3 - 9)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R ,
+--R radmult= 1,
+--R
+--R radvect =
+--R [
+--R [
+--R [
+--R +-+
+--R (- 36c + 36a b + 36)\|3
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c -
32a b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R +
+--R +---+
+--R (- 9\|- 3 - 9)
+--R *
+--R ROOT
+--R 4 3 2 2
2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b - 1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 2 +---+ 2
+--R (- 3c + (21a b + 9)c - 15a b - 6)\|- 3 - 3c + (21a b
+ 9)c
+--R +
+--R - 15a b - 6
+--R *
+--R +-+
+--R \|3
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R +
+--R +---+
+--R ((- 3c + 6)\|- 3 + 3c - 6)
+--R *
+--R ROOT
+--R 4 3 2 2 2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b - 1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 3 2 2 2
+---+
+--R (c + (11a b + 1)c + (- 11a b - 4)c + 24a b + 2a b + 2)\|-
3
+--R +
+--R 3 2 2 2
+--R - c + (- 11a b - 1)c + (11a b + 4)c - 24a b - 2a b - 2
+--R *
+--R +-+
+--R \|3
+--R /
+--R 2 +-+
+--R 108b \|3
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R ]
+--R ,
+--R
+--R [
+--R +---+
+--R (- 9\|- 3 - 9)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c -
32a b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R +
+--R +---+
+--R ((- 6c + 3)\|- 3 + 6c - 3)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R +
+--R 2
+--R 2c - 2c + 12a b + 2
+--R /
+--R +---+
+--R (9b\|- 3 - 9b)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3 2 2
2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b -
1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R ]
+--R ,
+--R [1]]
+--R ]
+--R ]
+--R ,
+--R
+--R [
+--R radval =
+--R +---+
+--R (- 9\|- 3 + 9)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c
+--R +
+--R 3 3 2 2
+--R - 32a b - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R +
+--R +---+
+--R ((3c + 3)\|- 3 + 3c + 3)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c -
32a b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R +
+--R 2
+--R - 2c + 2c - 12a b - 2
+--R /
+--R +---+
+--R (9\|- 3 + 9)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R ,
+--R radmult= 1,
+--R
+--R radvect =
+--R [
+--R [
+--R [
+--R +-+
+--R (- 36c + 36a b + 36)\|3
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c -
32a b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R +
+--R +---+
+--R (9\|- 3 - 9)
+--R *
+--R ROOT
+--R 4 3 2 2
2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b - 1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 2 +---+ 2
+--R (3c + (- 21a b - 9)c + 15a b + 6)\|- 3 - 3c + (21a b
+ 9)c
+--R +
+--R - 15a b - 6
+--R *
+--R +-+
+--R \|3
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R +
+--R +---+
+--R ((3c - 6)\|- 3 + 3c - 6)
+--R *
+--R ROOT
+--R 4 3 2 2 2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b - 1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 3 2 2 2
+---+
+--R (- c + (- 11a b - 1)c + (11a b + 4)c - 24a b - 2a b -
2)\|- 3
+--R +
+--R 3 2 2 2
+--R - c + (- 11a b - 1)c + (11a b + 4)c - 24a b - 2a b - 2
+--R *
+--R +-+
+--R \|3
+--R /
+--R 2 +-+
+--R 108b \|3
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R ]
+--R ,
+--R
+--R [
+--R +---+
+--R (- 9\|- 3 + 9)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c -
32a b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R +
+--R +---+
+--R ((- 6c + 3)\|- 3 - 6c + 3)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R +
+--R 2
+--R - 2c + 2c - 12a b - 2
+--R /
+--R +---+
+--R (9b\|- 3 + 9b)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3 2 2
2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b -
1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R ]
+--R ,
+--R [1]]
+--R ]
+--R ]
+--R ]
+--RType: List Record(radval: Expression Integer,radmult: Integer,radvect: List
Matrix Expression Integer)
+--E
+
+@
+but this works
+<<*>>=
+
+--S 238
ll 1
+--R
+--R (3)
+--R [
+--R radval =
+--R 9
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c -
32a b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R +
+--R (3c + 3)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R +
+--R 2
+--R c - c + 6a b + 1
+--R /
+--R 9
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3 2 2
2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b -
1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R ,
+--R radmult= 1,
+--R
+--R radvect =
+--R [
+--R [
+--R [
+--R +-+
+--R (- 18c + 18a b + 18)\|3
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c -
32a b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R +
+--R 9
+--R *
+--R ROOT
+--R 4 3 2 2
2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b - 1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 2 +-+
+--R (3c + (- 21a b - 9)c + 15a b + 6)\|3
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R +
+--R (- 3c + 6)
+--R *
+--R ROOT
+--R 4 3 2 2 2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b - 1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 3 2 2 2 +-+
+--R (c + (11a b + 1)c + (- 11a b - 4)c + 24a b + 2a b + 2)\|3
+--R /
+--R 2 +-+
+--R 54b \|3
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R ]
+--R ,
+--R
+--R [
+--R 9
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c -
32a b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R +
+--R (- 6c + 3)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R +
+--R 2
+--R c - c + 6a b + 1
+--R /
+--R 9b
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3 2 2
2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b -
1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R ]
+--R ,
+--R [1]]
+--R ]
+--R ]
+--RType: Record(radval: Expression Integer,radmult: Integer,radvect: List
Matrix Expression Integer)
+--E
+
+--S 239
ll 2
+--R
+--R (4)
+--R [
+--R radval =
+--R +---+
+--R (- 9\|- 3 - 9)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c -
32a b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R +
+--R +---+
+--R ((3c + 3)\|- 3 - 3c - 3)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R +
+--R 2
+--R 2c - 2c + 12a b + 2
+--R /
+--R +---+
+--R (9\|- 3 - 9)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3 2 2
2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b -
1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R ,
+--R radmult= 1,
+--R
+--R radvect =
+--R [
+--R [
+--R [
+--R +-+
+--R (- 36c + 36a b + 36)\|3
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c -
32a b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R +
+--R +---+
+--R (- 9\|- 3 - 9)
+--R *
+--R ROOT
+--R 4 3 2 2
2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b - 1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 2 +---+ 2
+--R (- 3c + (21a b + 9)c - 15a b - 6)\|- 3 - 3c + (21a b
+ 9)c
+--R +
+--R - 15a b - 6
+--R *
+--R +-+
+--R \|3
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R +
+--R +---+
+--R ((- 3c + 6)\|- 3 + 3c - 6)
+--R *
+--R ROOT
+--R 4 3 2 2 2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b - 1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 3 2 2 2
+---+
+--R (c + (11a b + 1)c + (- 11a b - 4)c + 24a b + 2a b + 2)\|-
3
+--R +
+--R 3 2 2 2
+--R - c + (- 11a b - 1)c + (11a b + 4)c - 24a b - 2a b - 2
+--R *
+--R +-+
+--R \|3
+--R /
+--R 2 +-+
+--R 108b \|3
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R ]
+--R ,
+--R
+--R [
+--R +---+
+--R (- 9\|- 3 - 9)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c -
32a b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R +
+--R +---+
+--R ((- 6c + 3)\|- 3 + 6c - 3)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R +
+--R 2
+--R 2c - 2c + 12a b + 2
+--R /
+--R +---+
+--R (9b\|- 3 - 9b)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3 2 2
2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b -
1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R ]
+--R ,
+--R [1]]
+--R ]
+--R ]
+--RType: Record(radval: Expression Integer,radmult: Integer,radvect: List
Matrix Expression Integer)
+--E
+
+--S 240
ll 3
+--R
+--R (5)
+--R [
+--R radval =
+--R +---+
+--R (- 9\|- 3 + 9)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c -
32a b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R +
+--R +---+
+--R ((3c + 3)\|- 3 + 3c + 3)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R +
+--R 2
+--R - 2c + 2c - 12a b - 2
+--R /
+--R +---+
+--R (9\|- 3 + 9)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3 2 2
2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b -
1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R ,
+--R radmult= 1,
+--R
+--R radvect =
+--R [
+--R [
+--R [
+--R +-+
+--R (- 36c + 36a b + 36)\|3
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c -
32a b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R +
+--R +---+
+--R (9\|- 3 - 9)
+--R *
+--R ROOT
+--R 4 3 2 2
2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b - 1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 2 +---+ 2
+--R (3c + (- 21a b - 9)c + 15a b + 6)\|- 3 - 3c + (21a b
+ 9)c
+--R +
+--R - 15a b - 6
+--R *
+--R +-+
+--R \|3
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R +
+--R +---+
+--R ((3c - 6)\|- 3 + 3c - 6)
+--R *
+--R ROOT
+--R 4 3 2 2 2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b - 1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 3 2 2 2
+---+
+--R (- c + (- 11a b - 1)c + (11a b + 4)c - 24a b - 2a b -
2)\|- 3
+--R +
+--R 3 2 2 2
+--R - c + (- 11a b - 1)c + (11a b + 4)c - 24a b - 2a b - 2
+--R *
+--R +-+
+--R \|3
+--R /
+--R 2 +-+
+--R 108b \|3
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R ]
+--R ,
+--R
+--R [
+--R +---+
+--R (- 9\|- 3 + 9)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c -
32a b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R **
+--R 2
+--R +
+--R +---+
+--R ((- 6c + 3)\|- 3 - 6c + 3)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3
+--R (- 4a b - 1)c + (6a b + 2)c
+--R +
+--R 2 2 2 2 2
3 3
+--R (- 13a b - 4a b - 1)c + (22a b + 6a b)c - 32a
b
+--R +
+--R 2 2
+--R - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R +
+--R 2
+--R - 2c + 2c - 12a b - 2
+--R /
+--R +---+
+--R (9b\|- 3 + 9b)
+--R *
+--R ROOT
+--R 9
+--R *
+--R ROOT
+--R 4 3 2 2
2
+--R (- 4a b - 1)c + (6a b + 2)c + (- 13a b - 4a b -
1)c
+--R +
+--R 2 2 3 3 2 2
+--R (22a b + 6a b)c - 32a b - 13a b - 4a b
+--R +
+--R 3 2 +-+
+--R (2c - 3c + (- 9a b - 3)c - 9a b + 2)\|3
+--R /
+--R +-+
+--R 54\|3
+--R ,
+--R 3
+--R ]
+--R ,
+--R [1]]
+--R ]
+--R ]
+--RType: Record(radval: Expression Integer,radmult: Integer,radvect: List
Matrix Expression Integer)
+--E
-
--------------------------------------------------------------------
--- address@hidden/11/24/93 Manuel Bronstein
--------------------------------------------------------------------
+@
address@hidden/11/24/93 Manuel Bronstein
+\begin{verbatim}
--)clear all
--Here's a bug that appears with the )math option only.
--
@@ -1180,314 +5734,554 @@ ll 3
--
-- Internal system problem in function appInfix :
-- fewer than 2 arguments to an infix function
+\end{verbatim}
+
address@hidden/12/1/93 Don Coppersmith
--------------------------------------------------------------------
--- address@hidden/12/1/93 Don Coppersmith
--------------------------------------------------------------------
+Attached is the list of integers n such that 2**512-n is prime
+and n is between 0 and 5000:
+\begin{verbatim}
+ 4893,4653,4475,4005,3893,3669,3459,3143,2967,
+ 2807,2529,1827,1695,975,875,629,569
+\end{verbatim}
+
+It was gotten from Axiom by issuing the commands
+<<*>>=
)clear all
---Attached is the list of integers n such that 2**512-n is prime
---and n is between 0 and 5000:
---
--- 4893,4653,4475,4005,3893,3669,3459,3143,2967,
--- 2807,2529,1827,1695,975,875,629,569
---It was gotten from Axiom by issuing the commands
+--S 241
qrimes : Stream Integer := generate(nextPrime,2**512-5000)
+--R
+--R (1)
+--R [
+--R
13407807929942597099574024998205846127479365820592393377723561443721764030_
+--R
0735469768018742981669034276900318581864860508537538828119465699464336490_
+--R 06079096
+--R ,
+--R
+--R
13407807929942597099574024998205846127479365820592393377723561443721764030_
+--R
0735469768018742981669034276900318581864860508537538828119465699464336490_
+--R 06079203
+--R ,
+--R
+--R
13407807929942597099574024998205846127479365820592393377723561443721764030_
+--R
0735469768018742981669034276900318581864860508537538828119465699464336490_
+--R 06079443
+--R ,
+--R
+--R
13407807929942597099574024998205846127479365820592393377723561443721764030_
+--R
0735469768018742981669034276900318581864860508537538828119465699464336490_
+--R 06079621
+--R ,
+--R
+--R
13407807929942597099574024998205846127479365820592393377723561443721764030_
+--R
0735469768018742981669034276900318581864860508537538828119465699464336490_
+--R 06080091
+--R ,
+--R
+--R
13407807929942597099574024998205846127479365820592393377723561443721764030_
+--R
0735469768018742981669034276900318581864860508537538828119465699464336490_
+--R 06080203
+--R ,
+--R
+--R
13407807929942597099574024998205846127479365820592393377723561443721764030_
+--R
0735469768018742981669034276900318581864860508537538828119465699464336490_
+--R 06080427
+--R ,
+--R
+--R
13407807929942597099574024998205846127479365820592393377723561443721764030_
+--R
0735469768018742981669034276900318581864860508537538828119465699464336490_
+--R 06080637
+--R ,
+--R
+--R
13407807929942597099574024998205846127479365820592393377723561443721764030_
+--R
0735469768018742981669034276900318581864860508537538828119465699464336490_
+--R 06080953
+--R ,
+--R
+--R
13407807929942597099574024998205846127479365820592393377723561443721764030_
+--R
0735469768018742981669034276900318581864860508537538828119465699464336490_
+--R 06081129
+--R ,
+--R ...]
+--R Type: Stream
Integer
+--E
+
+--S 242
rrimes := [ 2**512-p for p in qrimes while p < 2**512 ]
+--R
+--R (2) [5000,4893,4653,4475,4005,3893,3669,3459,3143,2967,...]
+--R Type: Stream
Integer
+--E
+
+--S 243
srimes := complete rrimes
+--R
+--R (3) [5000,4893,4653,4475,4005,3893,3669,3459,3143,2967,...]
+--R Type: Stream
Integer
+--E
+
+--S 244
[srimes.i for i in [1..18]]
+--R
+--R (4) [[5000,4893,4653,4475,4005,3893,3669,3459,3143,2967,...]]
+--R Type: List Stream
Integer
+--E
+
+--S 245
[srimes.i for i in [10..18]]
+--R
+--R (5) [[2967,2807,2529,1827,1695,975,875,629,569]]
+--R Type: List Stream
Integer
+--E
+
+@
address@hidden/12/13/93 (Mike Richardson)
--------------------------------------------------------------------
--- address@hidden/12/13/93 (Mike Richardson)
--------------------------------------------------------------------
+This is fixed now. You get an Expression Float.
+
+I suspect that this is an example of the same thing - but it looks worse:
+\begin{verbatim}
+ (145) ->X := log(0.7*%i*x)
+
+ >> Error detected within library code:
+ negative root
+\end{verbatim}
+<<*>>=
)clear all
---
---This is fixed now. You get an Expression Float.
---
--->
---> I suspect that this is an example of the same thing - but it looks
---> worse:
--->
---> (145) ->X := log(0.7*%i*x)
--->
---> >> Error detected within library code:
---> negative root
--->
---> Mike
--->
-
-
--------------------------------------------------------------------
--- address@hidden/12/15/93 (Robert S. Sutor)
--------------------------------------------------------------------
-)clear all
---enter Integer in input box
---choose domains
---choose Description
---click on Convertible(String)
---
---page cannot be formatted
--------------------------------------------------------------------
--- address@hidden/15/93 (barry trager)
--------------------------------------------------------------------
+--S 246
+X := log(0.7*%i*x)
+--R
+--R (1) log(0.7 %i x)
+--R Type: Expression Complex
Float
+--E
+
+@
+
address@hidden/12/15/93 (Robert S. Sutor)
+\begin{itemize}
+\item enter Integer in input box
+\item choose domains
+\item choose Description
+\item click on Convertible(String)
+\end{itemize}
+page cannot be formatted
+
address@hidden/15/93 (barry trager)
+<<*>>=
)clear all
+
+--S 247
log2() --> Float
-exp1() --> DoubleFLoat ???
+--R
+--R
+--R (1) 0.6931471805 5994530942
+--R Type:
Float
+--E
+--S 248
+exp1() --> DoubleFLoat ???
+--R
+--R
+--R (2) 2.7182818284 590452354
+--R Type:
Float
+--E
--------------------------------------------------------------------
--- address@hidden/12/14/93 Claude Quitte
--------------------------------------------------------------------
+@
address@hidden/12/14/93 Claude Quitte
+
+There is a bug in the function \verb|delete_!|
+for the type List ; this function,
+has two arguments, a list L and an integer n : \verb|delete_!(L, n)|
+returns a list L with the n nth-item deleted, MODIFYING its argument L.
+
+But for n = 1, the argument L is not MODIFIED !!! (exactly for n =
+minIndex(L)). An extract of the package ListAggregate (abbreviation LSAGG,
+file aggcat.spad).
+
+\begin{verbatim}
+ListAggregate(S:Object): Category == Join(StreamAggregate S,
+ FiniteLinearAggregate S, ExtensibleLinearAggregate S) with
+ list: S -> $
+ ++ list(x) returns the list of one element x.
+ add
+ ..........
+
+ delete_!(x:$, i:I) ==
+ i < (m := minIndex x) => error "index out of range"
+ i = m => rest x ****** <- here is the error
+ y := rest(x, (i - 1 - m)::N)
+ setrest_!(y, rest(y, 2))
+ x
+
+A little session :
+\end{verbatim}
+<<*>>=
)clear all
---There is a bug in the function delete_! for the type List ; this function,
---has two arguments, a list L and an integer n : delete_!(L, n) returns
---a list L with the n nth-item deleted, MODIFYING its argument L.
---
---But for n = 1, the argument L is not MODIFIED !!! (exactly for n =
---minIndex(L)). An extract of the package ListAggregate (abbreviation LSAGG,
---file aggcat.spad).
---
---
-----------------------------------------------------------------------
---ListAggregate(S:Object): Category == Join(StreamAggregate S,
--- FiniteLinearAggregate S, ExtensibleLinearAggregate S) with
--- list: S -> $
--- ++ list(x) returns the list of one element x.
--- add
--- ..........
---
--- delete_!(x:$, i:I) ==
--- i < (m := minIndex x) => error "index out of range"
--- i = m => rest x ****** <- here is the error
--- y := rest(x, (i - 1 - m)::N)
--- setrest_!(y, rest(y, 2))
--- x
---
---A little session :
---
+
+--S 249
L : List(String) := ["There is", "it seems", "a real bug", "here"]
--- (1) ["There is","it seems","a real bug","here"]
--- Type: List String
---
+--R
+--R (1) ["There is","it seems","a real bug","here"]
+--R Type: List
String
+--E
+
+--S 250
L1 := delete!(L, 1)
--- (2) ["it seems","a real bug","here"]
--- Type: List String
+--R
+--R (2) ["it seems","a real bug","here"]
+--R Type: List
String
+--E
+
+--S 251
L
--- (3) ["There is","it seems","a real bug","here"]
--- Type: List String
---
+--R
+--R (3) ["There is","it seems","a real bug","here"]
+--R Type: List
String
+--E
+
+--S 252
L2 := delete!(L, 2)
--- (4) ["There is","a real bug","here"]
--- Type: List String
+--R
+--R (4) ["There is","a real bug","here"]
+--R Type: List
String
+--E
+
+--S 253
L
--- (5) ["There is","a real bug","here"]
---
---Is there a way to fix this bug quickly ?? (I am ready to modify the source
---Axiom aggcat.spad).
---The problem with ! functions lies in the documentation not in the code. We
---need to explain more clearly that ! functions are ALLOWED to update their
---arguments, not REQUIRED to do so.
---Whenever using ! functions you should always
---use the returned result, not try to use the updated argument. This is how
---the code is designed.
---For example if you try to do insert! into an empty list,
---the argument is not updatable and thus a new list is created. The reason for
---having ! operations is for efficiency to do less copying, not to guarantee
---that given arguments will always be updated. So the correct use is to only
---depend on the results returned by updating functions.
-
-
--------------------------------------------------------------------
--- address@hidden/1/13/94 Claude Quitte
--------------------------------------------------------------------
+--R
+--R (5) ["There is","a real bug","here"]
+--R Type: List
String
+--E
+
+@
+\begin{verbatim}
+ (5) ["There is","a real bug","here"]
+\end{verbatim}
+Is there a way to fix this bug quickly ?? (I am ready to modify the source
+Axiom aggcat.spad).
+
+The problem with ! functions lies in the documentation not in the code. We
+need to explain more clearly that ! functions are ALLOWED to update their
+arguments, not REQUIRED to do so.
+
+Whenever using ! functions you should always
+use the returned result, not try to use the updated argument. This is how
+the code is designed.
+
+For example if you try to do insert! into an empty list,
+the argument is not updatable and thus a new list is created. The reason for
+having ! operations is for efficiency to do less copying, not to guarantee
+that given arguments will always be updated. So the correct use is to only
+depend on the results returned by updating functions.
+
address@hidden/1/13/94 Claude Quitte
+<<*>>=
)clear all
+
+--S 254
K := Fraction(Integer)
+--R
+--R (1) Fraction Integer
+--R Type:
Domain
+--E
+
+--S 255
PolK := UP('X, K)
+--R
+--R (2) UnivariatePolynomial(X,Fraction Integer)
+--R Type:
Domain
+--E
+
+--S 256
X : PolK := monomial(1, 1)
+--R
+--R (3) X
+--R Type: UnivariatePolynomial(X,Fraction
Integer)
+--E
+
+--S 257
n : PositiveInteger := 15
+--R
+--R (4) 15
+--R Type:
PositiveInteger
+--E
+
+--S 258
E := SimpleAlgebraicExtension(K, PolK, X**n + X**(n-3) -1)
+--R
+--R (5)
+--R SimpleAlgebraicExtension(Fraction Integer,UnivariatePolynomial(X,Fraction
Int
+--R eger),X**15+X**12-1)
+--R Type:
Domain
+--E
+
+--S 259
y : E := X::E
+--R
+--R (6) X
+--RType: SimpleAlgebraicExtension(Fraction
Integer,UnivariatePolynomial(X,Fraction Integer),X**15+X**12-1)
+--E
+
+--S 260
minimalPolynomial(y)$E
---
---Internal Error
---The function minimalPolynomial with signature SimpleAlgebraicExtension(
---Fraction Integer,UnivariatePolynomial(X,Fraction Integer),X**15+X**12-1)
----> UnivariatePolynomial(X,Fraction Integer) is missing from domain
---SimpleAlgebraicExtension(Fraction (Integer))
---(UnivariatePolynomial X (Fraction (Integer)))((15 1 . 1) (12 1 . 1) (0 -1 .
1))
+--R
+--R 15 12
+--R (7) X + X - 1
+--R Type: UnivariatePolynomial(X,Fraction
Integer)
+--E
+
+@
+\begin{verbatim}
+Internal Error
+The function minimalPolynomial with signature SimpleAlgebraicExtension(
+Fraction Integer,UnivariatePolynomial(X,Fraction Integer),X**15+X**12-1)
+ UnivariatePolynomial(X,Fraction Integer) is missing from domain
+SimpleAlgebraicExtension(Fraction (Integer))
+(UnivariatePolynomial X (Fraction (Integer)))((15 1 . 1) (12 1 . 1) (0 -1 . 1))
+\end{verbatim}
--------------------------------------------------------------------
--- address@hidden/02/20/94 (Richard Luczak)
--------------------------------------------------------------------
address@hidden/02/20/94 (Richard Luczak)
+
+I'm trying to define the following rule
+
+<<*>>=
)clear all
---
---I'm trying to define the following rule
---
+
+--S 261
tr := rule cos(x)**(n | integer? n and even? n)==(1-sin(x)**2)**(n/2)
---
---Axiom returns
---
--- Cannot find a definition or library operation named even? with argument
--- types
--- Expression Integer
---
--- We will attempt to step through and interpret the code.
---
--- n
--- -
--- n 2 2
--- (2) cos(x) == (- sin(x) + 1)
--- Type: RewriteRule(Integer,Integer,Expression Integer)
---
---
---and henceforth the rule does not work as desired. Variations on the same bug
---occur involving positive?. Of course, I get what I want using integer?(n/2)
---but ... Is this a bug?
---
--- from barry:
---
---Should use integer? n and even? integer n instead.
---
---I toyed with the idea of implementing:
--- even? x == integer? x and even? integer x
--- odd? x == integer? x and odd? integer x
---in EXPR, but this would mean that (x +-> even? x or odd? x) is not always
---true anymore and I'm not sure it's a good idea. If you guys don't think it's
---a problem, it takes 2 minutes to add it to EXPR R when R has RETRACT INT
---(don't use integer? but address@hidden(Integer, "failed")).
-
--------------------------------------------------------------------
--- address@hidden/02/22/94 don coppersmith
--------------------------------------------------------------------
-)clear all
---Questions that came up using Axiom on Aixproj:
---
---How does one find a Pade approximation? I tried pade(2,2,y)
---pade(y,2,2) Pade(2,2,y) PadeApproximation(2,2,y)
---(where y is a series in x) to no avail.
---
---Complaint: The error message
---
--- Could not find library function "pade" with arguments
--- (integer, integer, series).
---
---is misleading. If it can't find any "pade", then say
---
--- Could not find library function "pade".
---
---If there is such a function, but I've got the arguments wrong,
---why not say
---
--- The library function "pade" takes arguments
--- (series, integer, integer)
---
---or at least make that information available somehow.
---
--- from bob sutor:
---
---Don,
--- Try using )what op pade to see what is available. This will give
---you all operations with "pade" in their names. You can also search for
---*pade* via the HyperDoc Browse utility. If you don't use Browse, following
---up with, say, )display op pade, will give you a more information, though
---it is badly formatted (I will look at that).
---
---Aixproj is back-level, so I'm not sure exactly what is up their
---relative to the current system.
---
---I agree we could use better messages regarding unknown functions
---and I will look at that.
--- Bob
---
--- from bob sutor:
---
---The pade operation requires a taylor series as its third object while
---"series" returns a UnivariatePuiseuxSeries. If you set
--- y := taylor(1+x)**(1/5)
---then things work. You can also do
--- pade(1,1,y :: UTS(EXPR INT, x, 0) )
---but this is more work if you really have a Taylor series.
---
---This is unintuitive and there should be a "pade" for Puiseux series that
---fails if the series is not appropriate. Barry? I will now look at adding
---a bit more coercion support to the interpreter to allow Puiseux series
---to possibly retract to Laurent series and then possibly to Taylor series.
---
---Don,
--- I added the retractions I mentioned before to the development system.
---When the system is unsuccessful in matching a UnivariatePuiseuxSeries
---to an operation, it will first try to retract it to a
---UnivariateLaurentSeries. If that fails still, it will try to retract to
---a UnivariateTaylorSeries.
---
--- Be aware that this sort of thing happens when you are evaluating
---an expression. If you are defining a function, you should add
---explicit coercions or create objects of the correct type for
---efficiency. You function may still work, but it may not run in
---a compiled form.
+--R
+--R n
+--R -
+--R n 2 2
+--R (1) cos(x) == (- sin(x) + 1)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
--------------------------------------------------------------------
--- address@hidden/02/24/94 Barry Trager
--------------------------------------------------------------------
+@
+Axiom returns
+\begin{verbatim}
+ Cannot find a definition or library operation named even? with argument
+ types
+ Expression Integer
+ We will attempt to step through and interpret the code.
+ n
+ -
+ n 2 2
+ (2) cos(x) == (- sin(x) + 1)
+ Type: RewriteRule(Integer,Integer,Expression Integer)
+
+\end{verbatim}
+and henceforth the rule does not work as desired. Variations on the same bug
+occur involving positive?. Of course, I get what I want using integer?(n/2)
+but ... Is this a bug?
+
+from barry:
+
+Should use integer? n and even? integer n instead.
+
+I toyed with the idea of implementing:
+\begin{verbatim}
+ even? x == integer? x and even? integer x
+ odd? x == integer? x and odd? integer x
+\end{verbatim}
+in EXPR, but this would mean that (x +-> even? x or odd? x) is not always
+true anymore and I'm not sure it's a good idea. If you guys don't think it's
+a problem, it takes 2 minutes to add it to EXPR R when R has RETRACT INT
+(don't use integer? but address@hidden(Integer, "failed")).
+
address@hidden/02/22/94 don coppersmith
+
+Questions that came up using Axiom on Aixproj:
+
+How does one find a Pade approximation? I tried pade(2,2,y)
+pade(y,2,2) Pade(2,2,y) PadeApproximation(2,2,y)
+(where y is a series in x) to no avail.
+
+Complaint: The error message
+\begin{verbatim}
+ Could not find library function "pade" with arguments
+ (integer, integer, series).
+\end{verbatim}
+is misleading. If it can't find any "pade", then say
+\begin{verbatim}
+ Could not find library function "pade".
+\end{verbatim}
+If there is such a function, but I've got the arguments wrong,
+why not say
+\begin{verbatim}
+ The library function "pade" takes arguments
+ (series, integer, integer)
+\end{verbatim}
+or at least make that information available somehow.
+
+from bob sutor:
+
+Don,
+
+Try using )what op pade to see what is available. This will give
+you all operations with "pade" in their names. You can also search for
+*pade* via the HyperDoc Browse utility. If you don't use Browse, following
+up with, say, )display op pade, will give you a more information, though
+it is badly formatted (I will look at that).
+
+Aixproj is back-level, so I'm not sure exactly what is up their
+relative to the current system.
+
+I agree we could use better messages regarding unknown functions
+and I will look at that.
+
+Bob
+
+from bob sutor:
+
+The pade operation requires a taylor series as its third object while
+``series'' returns a UnivariatePuiseuxSeries. If you set
+\begin{verbatim}
+ y := taylor(1+x)**(1/5)
+\end{verbatim}
+then things work. You can also do
+\begin{verbatim}
+ pade(1,1,y :: UTS(EXPR INT, x, 0) )
+\end{verbatim}
+but this is more work if you really have a Taylor series.
+
+This is unintuitive and there should be a ``pade'' for Puiseux series that
+fails if the series is not appropriate. Barry? I will now look at adding
+a bit more coercion support to the interpreter to allow Puiseux series
+to possibly retract to Laurent series and then possibly to Taylor series.
+
+Don,
+
+I added the retractions I mentioned before to the development system.
+When the system is unsuccessful in matching a UnivariatePuiseuxSeries
+to an operation, it will first try to retract it to a
+UnivariateLaurentSeries. If that fails still, it will try to retract to
+a UnivariateTaylorSeries.
+
+Be aware that this sort of thing happens when you are evaluating
+an expression. If you are defining a function, you should add
+explicit coercions or create objects of the correct type for
+efficiency. You function may still work, but it may not run in
+a compiled form.
+
address@hidden/02/24/94 Barry Trager
+<<*>>=
)clear all
+
+--S 262
sqrt(2)*2.0 -- fails
+--R
+--R
+--R (1) 2.8284271247 461900976
+--R Type: Expression
Float
+--E
+
+--S 263
sqrt(2)::EXPR INT * 2.0 -- works
+--R
+--R
+--R (2) 2.8284271247 461900976
+--R Type: Expression
Float
+--E
--- tpd: this now fails with sample : () -> Reference Integer is missing
--- from domain: Integer
--- Internal Error
+@
+tpd: this now fails with sample : () -> Reference Integer is missing
+from domain: Integer
+Internal Error
address@hidden/02/28/94 Manuel Bronstein
--------------------------------------------------------------------
--- address@hidden/02/28/94 Manuel Bronstein
--------------------------------------------------------------------
+<<*>>=
)clear all
+
+--S 264
f := exp(exp(x)*exp(1/exp(x))) / exp exp x
---)tr LIMITPS )math
---)tr DFINTTLS )math
---)tr DFINTTLS_;findLimit
--- the following exits back to the shell after limit returns "failed":
-computeInt(first tower f, f, 0, %plusInfinity, false)$DFINTTLS(INT, EXPR INT)
-
--------------------------------------------------------------------
--- address@hidden/03/02/94 James Davenport
--------------------------------------------------------------------
-)clear all
---I've fixed coercions of List and Vector to DirectProduct.
---You can now do things like
+--R
+--R 1
+--R ---
+--R x
+--R x %e
+--R %e %e
+--R %e
+--R (1) ----------
+--R x
+--R %e
+--R %e
+--R Type: Expression
Integer
+--E
+
+@
+\begin{verbatim}
+ )tr LIMITPS )math
+ )tr DFINTTLS )math
+ )tr DFINTTLS_;findLimit
+\end{verbatim}
+the following exits back to the shell after limit returns ``failed'':
+computeInt(first tower f, f, 0, \%plusInfinity, false)\$DFINTTLS(INT, EXPR INT)
+
address@hidden/03/02/94 James Davenport
+
+I've fixed coercions of List and Vector to DirectProduct.
+You can now do things like
+
+<<*>>=
+)clear all
+
+--S 265
[1,2,3] :: DirectProduct(3, Fraction Integer)
---
---I also fixed coercions from Vector to List, which seems to have
---been removed via a library change.
+--R
+--R (1) [1,2,3]
+--R Type: DirectProduct(3,Fraction
Integer)
+--E
+
+@
+I also fixed coercions from Vector to List, which seems to have
+been removed via a library change.
+
address@hidden/03/09/94 James Davenport
--------------------------------------------------------------------
--- address@hidden/03/09/94 James Davenport
--------------------------------------------------------------------
+Used to give a lisp error
+<<*>>=
)clear all
--- used to give a lisp error
+
+--S 266
x**10+1::Polynomial PrimeField 2
+--R
+--R 10
+--R (1) x + 1
+--R Type: Polynomial PrimeField
2
+--E
--------------------------------------------------------------------
--- address@hidden/03/11/94 Themos Tsikas
--------------------------------------------------------------------
+@
address@hidden/03/11/94 Themos Tsikas
+<<*>>=
)clear all
+
+--S 267
f(x)==x**2
+--R Type:
Void
+--E
+
+@
draw(f,-1..1)
--------------------------------------------------------------------
--- address@hidden/03/16/94 Themos Tsikas
--------------------------------------------------------------------
-)clear all
---The problem was that nobody was around to catch the coerceOrCroaker.
---Just comment out the first line of intCodeGenCoerce1 in i-code.boot
---(the one with the explicit THROW). Bothe the riemannSphereDraw and JHD's
---original example will then work. -- bob sutor
address@hidden/03/16/94 Themos Tsikas
+
+The problem was that nobody was around to catch the coerceOrCroaker.
+Just comment out the first line of intCodeGenCoerce1 in i-code.boot
+(the one with the explicit THROW). Bothe the riemannSphereDraw and JHD's
+original example will then work. -- bob sutor
+
+\begin{verbatim}
)read conformal
riemannSphereDraw(1..2,0..%pi,4,4,"polar")
+\end{verbatim}
-
+<<*>>=
+)spool
+)lisp (bye)
@
\eject
diff --git a/src/input/schaum17.input.pamphlet
b/src/input/schaum17.input.pamphlet
index 6c5e67c..fdc11a2 100644
--- a/src/input/schaum17.input.pamphlet
+++ b/src/input/schaum17.input.pamphlet
@@ -776,7 +776,7 @@ bb:=-1/a*tan(%pi/4-(a*x)/2)
--S 57 of 136
cc:=aa-bb
---R
+--R
--R
--R 2a x - %pi
--R (- sin(a x) - cos(a x) - 1)tan(----------) - 2cos(a x) - 2
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