Re: Generic Givens Rotation Matrix in EQN

[Damian McGuckin]

On Mon, 1 Mar 2021, Tadziu Hoffmann wrote:

> This is fairly straightforward with a few suitable eqn definitions to 
> simplify entering the equation.

> The attached PDF was created with the following code and my own 
> paragraph and section macros.  (Use "left [" and "right ]" instead of my 
> bracket macro if you want to use the equation with some other macro 
> package.)

Done. Works flawlessly.  See below.

> Note that since I've entered the matrix column-wise it appears 
> transposed in the text editor display.

Yes.

>  define ...   % roman " . . . " %
>  define vdots % roman "\v'-.6m'\z.\v'.5m'\z.\v'.5m'.\v'-.4m'" %
>  define ddots % roman "\v'-.6m' .\v'.5m' .\v'.5m' .\v'-.4m' " %

Exactly what I needed. I was having brain fade trying to do this.

Thanks heaps

.\" Begin example using -MM
.S 12 14
.EQ
define ...   % roman " . . . " %
define vdots % roman "\v'-.6m'\z.\v'.5m'\z.\v'.5m'.\v'-.4m'" %
define ddots % roman "\v'-.6m' .\v'.5m' .\v'.5m' .\v'-.4m' " %
define Gjj % G sub back 30 jj %
define Gji % G sub back 30 ji %
define Gii % G sub back 10 ii %
define Gij % G sub back 10 ij %
define Gkk % G sub back 10 kk %
define | % above %
delim $$
.EN
.DS 1
.EQ
set column_sep 20
G ( i, j, theta ) ~~ = ~~
left [
matrix {
ccol { 1   | vdots | 0   | vdots | 0   | vdots | 0   }
ccol { ... | ddots | ... | ""    | ... | ""    | ... }
ccol { 0   | vdots | c   | vdots | s   | vdots | 0   }
ccol { ... | ""    | ... | ddots | ... | ""    | ... }
ccol { 0   | vdots | -s  | vdots | c   | vdots | 0   }
ccol { ... | ""    | ... | ""    | ... | ddots | ... }
ccol { 0   | vdots | 0   | vdots | 0   | vdots | 1   }
}
right ]
.EN
.DE
which is, when $i > j$,
.DS 1
.EQ
set column_sep 100
Gkk = 1 roman "  for  " k != i, j , fwd 100
Gii =   Gjj = + c = + cos ( theta )     , fwd 100
Gji = - Gij = - s = - sin ( theta ) .
.EN
.DE
(See https://en.wikipedia.org/wiki/Givens_rotation.)
.\" End of example using -MM

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