f5fa93572d
* Gram-Schmidt using vector type * QR decomposition * Operator 1-norm and maximum norm; stub for 2-norm and angle between subspaces (`matrix-basis-angle') * `matrix-absolute-error' and `matrix-relative-error'; also predicates based on them, such as `matrix-identity?' * Lots of shuffling code about * Types that can have contracts, and an exhaustive test to make sure every value exported by `math/matrix' has a contract when used in untyped code * Some more tests (still needs some)
40 lines
1.5 KiB
Racket
40 lines
1.5 KiB
Racket
#lang typed/racket/base
|
|
|
|
(require "matrix-types.rkt"
|
|
"matrix-basic.rkt"
|
|
"matrix-arithmetic.rkt"
|
|
"matrix-constructors.rkt"
|
|
"matrix-gram-schmidt.rkt"
|
|
"../array/array-transform.rkt")
|
|
|
|
(provide matrix-qr)
|
|
|
|
#|
|
|
QR decomposition currently does Gram-Schmidt twice, as suggested by
|
|
|
|
Luc Giraud, Julien Langou, Miroslav Rozloznik.
|
|
On the round-off error analysis of the Gram-Schmidt algorithm with reorthogonalization.
|
|
Technical Report, 2002.
|
|
|
|
It normalizes only the second time.
|
|
|
|
I've verified experimentally that, with random, square matrices (elements in [0,1]), doing so
|
|
produces matrices for which `matrix-orthogonal?' returns #t with eps <= 10*epsilon.0, apparently
|
|
independently of the matrix size.
|
|
|#
|
|
|
|
(: matrix-qr (case-> ((Matrix Real) -> (Values (Matrix Real) (Matrix Real)))
|
|
((Matrix Real) Any -> (Values (Matrix Real) (Matrix Real)))
|
|
((Matrix Number) -> (Values (Matrix Number) (Matrix Number)))
|
|
((Matrix Number) Any -> (Values (Matrix Number) (Matrix Number)))))
|
|
(define (matrix-qr M [full? #t])
|
|
(define B (matrix-gram-schmidt M #f))
|
|
(define Q
|
|
(matrix-gram-schmidt
|
|
(cond [(or (square-matrix? B) (and (matrix? B) (not full?))) B]
|
|
[(matrix? B) (array-append* (list B (matrix-basis-extension B)) 1)]
|
|
[full? (identity-matrix (matrix-num-rows M))]
|
|
[else (matrix-col (identity-matrix (matrix-num-rows M)) 0)])
|
|
#t))
|
|
(values Q (matrix-upper-triangle (matrix* (matrix-hermitian Q) M))))
|