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Documentation on basic matrix operations and matrix-qr.

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Jens Axel Søgaard 2013-01-02 19:25:04 +01:00
parent 18389532e7
commit 69406bf156
1 changed files with 49 additions and 13 deletions
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collects/math/scribblings/math-matrix.scrbl
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@ -409,7 +409,7 @@ The function @racket[matrix-map-cols] works likewise but on rows.
@examples[#:eval untyped-eval
(define A (matrix ([1 2 3] [4 5 6] [7 8 9] [10 11 12])))
(define (double-row r) (matrix-scale r 2))
(matrix-map-rows double-row A)]
(matrix-map-rows double-row A)]
}
@ -418,23 +418,34 @@ The function @racket[matrix-map-cols] works likewise but on rows.
@section[#:tag "matrix:basic"]{Basic Operations}
@defthing[matrix-conjugate Procedure]{
@;{(: matrix-conjugate ((Matrix Number) -> (Matrix Number)))}
@defproc[(matrix-conjugate [M (Matrix A)]) (Matrix A)]{
Returns a matrix where each element of the given matrix is conjugated.
@examples[#:eval untyped-eval
(matrix-conjugate (matrix ([1 +i] [-1 2+i])))]
}
@deftogether[(@defthing[matrix-transpose Procedure]
@defthing[matrix-hermitian Procedure])]{
@;{
(: matrix-transpose (All (A) (Matrix A) -> (Matrix A)))
(: matrix-hermitian ((Matrix Number) -> (Matrix Number)))
}
@margin-note{@hyperlink["http://en.wikipedia.org/wiki/Transpose"]{Wikipedia: Transpose}}
@deftogether[(@defproc[(matrix-transpose [M (Matrix A)]) (Matrix A)]
@defproc[(matrix-hermitian [M (Matrix A)]) (Matrix A)])]{
@margin-note{@hyperlink["http://en.wikipedia.org/wiki/Hermitian_matrix"]{Wikipedia: Hermitian}}
Returns the transpose or the hermitian of the matrix.
The hermitian of a matrix is the conjugate of the transposed matrix.
For a real matrix these operations return the the same result.
@examples[#:eval untyped-eval
(matrix-transpose (matrix ([1 1] [2 2] [3 3])))
(matrix-hermitian (matrix ([1 +i] [2 +2i] [3 +3i])))]
}
@defthing[matrix-trace Procedure]{
@;{(: matrix-trace ((Matrix Number) -> Number))}
@margin-note{@hyperlink["http://en.wikipedia.org/wiki/Trace_(linear_algebra)"]{Wikipedia: Trace}}
@defproc[(matrix-trace [M (Matrix Number)]) (Matrix Number)]{
Returns the trace of the square matrix. The trace of matrix is the
the sum of the diagonal elements.
@examples[#:eval untyped-eval
(matrix-trace (matrix ([1 2] [3 4])))]
}
@;{==================================================================================================}
@ -555,9 +566,34 @@ Equivalent to @racket[(acos (matrix-cos-angle M0 M1))].
@defthing[matrix-basis-extension Procedure]{}
@defthing[matrix-qr Procedure]{}
@margin-note{@hyperlink["http://en.wikipedia.org/wiki/QR_decomposition"]{Wikipedia: QR decomposition}}
@deftogether[(@defproc[(matrix-qr [M (Matrix Real)]) (Values (Matrix Real) (Matrix Real))]
@defproc[(matrix-qr [M (Matrix Real)] [full Any]) (Values (Matrix Real) (Matrix Real))]
@defproc[(matrix-qr [M (Matrix Number)]) (Values (Matrix Number) (Matrix Number))]
@defproc[(matrix-qr [M (Matrix Number)] [full Any]) (Values (Matrix Number) (Matrix Number))])]{
Computes a QR-decomposition of the matrix @racket[M]. The values returned are
the matrices @racket[Q] and @racket[R]. If @racket[full] is false, then
a reduced decomposition is returned, otherwise a full decomposition is returned.
@margin-note{An @italic{orthonormal} matrix has columns which are orthooginal, unit vectors.}
The (full) decomposition of a square matrix consists of two matrices:
a orthogonal matrix @racket[Q] and an upper triangular matrix @racket[R],
such that @racket[QR = M].
For tall non-square matrices @racket[R], the triangular part of the full decomposition,
contains zeros below the diagonal. The reduced decomposition leaves the zeros out.
See the Wikipedia entry on @hyperlink["http://en.wikipedia.org/wiki/QR_decomposition"]{QR decomposition}
for more details.
The decomposition @racket[M = QR] is useful for solving the equation @racket[Mx=v].
Since the inverse of Q is simply the transpose of Q,
@racket[Mx=v <=> QRx=v <=> Rx = Q^T v].
And since @racket[R] is upper triangular, the system can be solved by back substitution.
The algorithm used is Gram-Schmidt with reorthogonalization.
See the paper @hyperlink["http://www.cerfacs.fr/algor/reports/2002/TR_PA_02_33.pdf"]{On the round-off error analysis of the Gram-Schmidt algorithm with reorthogonalization.}
by Luc Giraud, Julien Langou, Miroslav Rozloznik.
}
@;{==================================================================================================}