e5eb9751f0
* At least stubbed out all (or almost all) `math/matrix' exports; many have complete documentation (e.g. types, predicates, accessors, constructors, for loops, conversion, much of "Comparing Matrices") * Moved `matrix-zero?' and fixed to use (matrix-error-norm) as its norm * Added `matrix-basis-cos-angle' (currently a stub; should return smallest singular value of a certain matrix multiplication)
621 lines
24 KiB
Racket
621 lines
24 KiB
Racket
#lang scribble/manual
|
||
|
||
@(require scribble/eval
|
||
racket/sandbox
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||
(for-label racket/base racket/vector racket/match racket/unsafe/ops racket/string
|
||
racket/list
|
||
math plot
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||
(only-in typed/racket/base
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||
ann inst : λ: define: make-predicate ->
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||
Flonum Real Boolean Any Integer Index Natural Exact-Positive-Integer
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||
Nonnegative-Real Sequenceof Fixnum Values Number Float-Complex
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||
All U List Vector Listof Vectorof Struct FlVector
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||
Symbol Output-Port))
|
||
"utils.rkt")
|
||
|
||
@(define untyped-eval (make-untyped-math-eval))
|
||
@interaction-eval[#:eval untyped-eval
|
||
(require racket/match
|
||
racket/vector
|
||
racket/string
|
||
racket/sequence
|
||
racket/list)]
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||
|
||
@(define typed-eval (make-math-eval))
|
||
@interaction-eval[#:eval typed-eval
|
||
(require racket/match
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||
racket/vector
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||
racket/string
|
||
racket/sequence
|
||
racket/list)]
|
||
|
||
@title[#:tag "matrices" #:style 'toc]{Matrices and Linear Algebra}
|
||
@(author-jens-axel)
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||
@(author-neil)
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||
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||
@bold{Performance Warning:} Matrix values are arrays, as exported by @racketmodname[math/array].
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||
The same performance warning applies: operations are currently 25-50 times slower in untyped Racket
|
||
than in Typed Racket, due to the overhead of checking higher-order contracts. We are working on it.
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||
|
||
For now, if you need speed, use the @racketmodname[typed/racket] language.
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||
|
||
@defmodule[math/matrix]
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||
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||
Documentation for this module is currently under construction.
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||
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||
Intro topics: definitions, case-> types, non-strictness
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||
|
||
@local-table-of-contents[]
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||
|
||
|
||
@;{==================================================================================================}
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||
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||
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||
@section[#:tag "matrix:types"]{Types, Predicates and Accessors}
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||
|
||
@defform[(Matrix A)]{
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||
Equivalent to @racket[(Array A)], but used for values @racket[M] for which @racket[(matrix? M)] is
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||
@racket[#t].
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||
}
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||
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||
@defproc[(matrix? [arr (Array A)]) Boolean]{
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||
Returns @racket[#t] when @racket[arr] is a @deftech{matrix}: a nonempty array with exactly two axes.
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||
@examples[#:eval typed-eval
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||
(matrix? (array 10))
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||
(matrix? (array #[1 2 3]))
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(matrix? (make-array #(5 0) 0))
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(matrix? (array #[#[1 0] #[0 1]]))]
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||
}
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||
|
||
@defproc[(row-matrix? [arr (Array A)]) Boolean]{
|
||
Returns @racket[#t] when @racket[arr] is a @deftech{row matrix}:
|
||
a @tech{matrix} with exactly one row.
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||
}
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||
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||
@defproc[(col-matrix? [arr (Array A)]) Boolean]{
|
||
Returns @racket[#t] when @racket[arr] is a @deftech{column matrix}:
|
||
a @tech{matrix} with exactly one column.
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||
}
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||
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||
@defproc[(square-matrix? [arr (Array A)]) Boolean]{
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||
Returns @racket[#t] when @racket[arr] is a @tech{matrix} with the same number of rows and columns.
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||
}
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||
|
||
@defproc[(matrix-shape [M (Matrix A)]) (Values Index Index)]{
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||
Returns @racket[M]'s row and column count, respectively.
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||
Raises an error if @racket[(matrix? M)] is @racket[#f].
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||
@examples[#:eval typed-eval
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||
(matrix-shape (row-matrix [1 2 3]))
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(matrix-shape (col-matrix [1 2 3]))
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(matrix-shape (identity-matrix 3))]
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||
}
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||
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||
@defproc[(matrix-num-rows [M (Matrix A)]) Index]{
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||
Returns the number of rows in @racket[M], or the first value of @racket[(matrix-shape M)].
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||
}
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||
|
||
@defproc[(matrix-num-cols [M (Matrix A)]) Index]{
|
||
Returns the number of columns in @racket[M], or the second value of @racket[(matrix-shape M)].
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||
}
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||
|
||
@defproc[(square-matrix-size [M (Matrix A)]) Index]{
|
||
Returns the number of rows/columns in @racket[M].
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||
Raises an error if @racket[(square-matrix? M)] is @racket[#f].
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@examples[#:eval typed-eval
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(square-matrix-size (identity-matrix 3))
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(square-matrix-size (row-matrix [1 2 3]))]
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}
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||
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||
|
||
@;{==================================================================================================}
|
||
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||
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||
@section[#:tag "matrix:construction"]{Construction}
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||
|
||
@defform/subs[(matrix [[expr ...+] ...+] maybe-type-ann)
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[(maybe-type-ann (code:line) (code:line : type))]]{
|
||
Like the @racket[array] form for creating arrays, but does not require @racket[#[...]] to delimit
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||
nested rows, and the result is constrained to be a @racket[matrix?].
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||
@examples[#:eval typed-eval
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(matrix [[1 2 3] [4 5 6]])
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(matrix [[1 2 3] [4 5 6]] : Number)
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(matrix [[]])]
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||
}
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||
|
||
@defform/subs[(row-matrix [expr ...+] maybe-type-ann)
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[(maybe-type-ann (code:line) (code:line : type))]]{
|
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Like @racket[matrix], but returns a @tech{row matrix}.
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||
@examples[#:eval typed-eval
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||
(row-matrix [1 2 3])
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||
(row-matrix [1 2 3] : Number)
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(row-matrix [])]
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||
}
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||
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||
@defform/subs[(col-matrix [expr ...+] maybe-type-ann)
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[(maybe-type-ann (code:line) (code:line : type))]]{
|
||
Like @racket[matrix], but returns a @tech{column matrix}.
|
||
@examples[#:eval typed-eval
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(col-matrix [1 2 3])
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(col-matrix [1 2 3] : Number)
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(col-matrix [])]
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}
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||
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@defproc[(identity-matrix [n Integer]) (Matrix (U 0 1))]{
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||
Returns an @racket[n]×@racket[n] identity matrix; @racket[n] must be positive.
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||
}
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||
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@defproc[(make-matrix [m Integer] [n Integer] [x A]) (Matrix A)]{
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Returns an @racket[m]×@racket[n] matrix filled with the value @racket[x];
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both @racket[m] and @racket[n] must be positive.
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Analogous to @racket[make-array] (and defined in terms of it).
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||
}
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@defproc[(build-matrix [m Integer] [n Integer] [proc (Index Index -> A)]) (Matrix A)]{
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Returns an @racket[m]×@racket[n] matrix with entries returned by @racket[proc];
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||
both @racket[m] and @racket[n] must be positive.
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||
Analogous to @racket[build-array] (and defined in terms of it).
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}
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||
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@defproc[(diagonal-matrix [xs (Listof A)]) (Matrix (U A 0))]{
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||
Returns a matrix with @racket[xs] along the diagonal and @racket[0] everywhere else.
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||
The length of @racket[xs] must be positive.
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}
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||
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@margin-note{@hyperlink["http://en.wikipedia.org/wiki/Block_matrix#Block_diagonal_matrices"]{Wikipedia: Block-diagonal matrices}}
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@defproc[(block-diagonal-matrix [Xs (Listof (Matrix A))]) (Matrix (U A 0))]{
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Returns a matrix with matrices @racket[Xs] along the diagonal and @racket[0] everywhere else.
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||
The length of @racket[Xs] must be positive.
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@examples[#:eval typed-eval
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(block-diagonal-matrix (list (matrix [[6 7] [8 9]])
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(diagonal-matrix '(7 5 7))
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(col-matrix [1 2 3])
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(row-matrix [4 5 6])))]
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}
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@margin-note{@hyperlink["http://en.wikipedia.org/wiki/Vandermonde_matrix"]{Wikipedia: Vandermonde matrix}}
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@defproc[(vandermonde-matrix [xs (Listof Number)] [n Integer]) (Matrix Number)]{
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Returns an @racket[m]×@racket[n] Vandermonde matrix, where @racket[m = (length xs)].
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@examples[#:eval typed-eval
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(vandermonde-matrix '(1 2 3 4) 5)]
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Using a Vandermonde matrix to find a Lagrange polynomial (the polynomial of least degree that passes
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||
through a given set of points):
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@interaction[#:eval untyped-eval
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(define (lagrange-polynomial xs ys)
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(array->list (matrix-solve (vandermonde-matrix xs (length xs))
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(->col-matrix ys))))
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(define xs '(-3 0 3))
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(define ys '(13 3 6))
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(match-define (list c b a) (lagrange-polynomial xs ys))
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(plot (list (function (λ (x) (+ c (* b x) (* a x x))) -4 4)
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(points (map list xs ys))))]
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Note that the above example is in untyped Racket.
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||
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||
This function is defined in terms of @racket[array-axis-expand].
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||
}
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@deftogether[(@defform[(for/matrix: m n maybe-fill (for:-clause ...) maybe-type-ann
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||
body ...+)]
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@defform/subs[(for*/matrix: m n maybe-fill (for:-clause ...) maybe-type-ann
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||
body ...+)
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||
([maybe-fill (code:line) (code:line #:fill fill)]
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||
[maybe-type-ann (code:line) (code:line : body-type)])
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||
#:contracts ([m Integer]
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||
[n Integer]
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[fill body-type])])]{
|
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Like @racket[for/array:] and @racket[for*/array:], but for matrices.
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||
The only material difference is that the shape @racket[m n] is required and must be positive.
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||
}
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||
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||
@deftogether[(@defform[(for/matrix m n maybe-fill (for-clause ...)
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||
body ...+)]
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||
@defform[(for*/matrix m n maybe-fill (for-clause ...)
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||
body ...+)])]{
|
||
Untyped versions of the loop macros.
|
||
}
|
||
|
||
|
||
@;{==================================================================================================}
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||
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||
|
||
@section[#:tag "matrix:conversion"]{Conversion}
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||
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||
@deftogether[(@defproc[(list->matrix [m Integer] [n Integer] [xs (Listof A)]) (Matrix A)]
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||
@defproc[(matrix->list [M (Matrix A)]) (Listof A)])]{
|
||
Convert a flat list to an @racket[m]×@racket[n] matrix and back;
|
||
both @racket[m] and @racket[n] must be positive, and @racket[(* m n) = (length xs)].
|
||
The entries in @racket[xs] are in row-major order.
|
||
@examples[#:eval typed-eval
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||
(list->matrix 2 3 '(1 2 3 4 5 6))
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||
(matrix->list (matrix [[1 2] [3 4] [5 6]]))]
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||
}
|
||
|
||
@deftogether[(@defproc[(vector->matrix [m Integer] [n Integer] [xs (Vectorof A)]) (Matrix A)]
|
||
@defproc[(matrix->vector [M (Matrix A)]) (Vectorof A)])]{
|
||
Like @racket[list->matrix] and @racket[matrix->list], but for vectors.
|
||
@examples[#:eval typed-eval
|
||
(vector->matrix 2 3 #(1 2 3 4 5 6))
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||
(matrix->vector (matrix [[1 2] [3 4] [5 6]]))]
|
||
}
|
||
|
||
@deftogether[(@defproc[(->row-matrix [xs (U (Listof A) (Vectorof A) (Array A))]) (Matrix A)]
|
||
@defproc[(->col-matrix [xs (U (Listof A) (Vectorof A) (Array A))]) (Matrix A)])]{
|
||
Convert a list, vector, or array into a row or column matrix.
|
||
If @racket[xs] is an array, it must be nonempty and @bold{not} have more than one axis with length
|
||
greater than @racket[1].
|
||
@examples[#:eval typed-eval
|
||
(->row-matrix '(1 2 3))
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||
(->row-matrix #(1 2 3))
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||
(->row-matrix (col-matrix [1 2 3]))
|
||
(->col-matrix (array #[#[#[1]] #[#[2]] #[#[3]]]))
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||
(->col-matrix (matrix [[1 0] [0 1]]))]
|
||
}
|
||
|
||
@deftogether[(@defproc[(list*->matrix [xss (Listof (Listof A))]) (Matrix A)]
|
||
@defproc[(matrix->list* [M (Matrix A)]) (Listof (Listof A))])]{
|
||
Convert a list of lists of entries into a matrix and back.
|
||
@examples[#:eval typed-eval
|
||
(list*->matrix '((1 2 3) (4 5 6)))
|
||
(matrix->list* (matrix [[1 2 3] [4 5 6]]))]
|
||
These functions are like @racket[list*->array] and @racket[array->list*], but use a fixed-depth
|
||
(i.e. non-recursive) list type, and do not require a predicate to distinguish entries from rows.
|
||
}
|
||
|
||
@deftogether[(@defproc[(vector*->matrix [xss (Vectorof (Vectorof A))]) (Matrix A)]
|
||
@defproc[(matrix->vector* [M (Matrix A)]) (Vectorof (Vectorof A))])]{
|
||
Like @racket[list*->matrix] and @racket[matrix*->list], but for vectors.
|
||
@examples[#:eval typed-eval
|
||
((inst vector*->matrix Integer) #(#(1 2 3) #(4 5 6)))
|
||
(matrix->vector* (matrix [[1 2 3] [4 5 6]]))]
|
||
As in the first example, Typed Racket often needs help inferring the type @racket[A].
|
||
}
|
||
|
||
|
||
@;{==================================================================================================}
|
||
|
||
|
||
@section[#:tag "matrix:arith"]{Entrywise Operations and Arithmetic}
|
||
|
||
TODO: fill in empty docs
|
||
|
||
@defproc*[([(matrix-map [f (A -> R)] [arr0 (Matrix A)]) (Matrix R)]
|
||
[(matrix-map [f (A B Ts ... -> R)] [arr0 (Matrix A)] [arr1 (Matrix B)] [arrs (Matrix Ts)]
|
||
...)
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||
(Matrix R)])]{
|
||
Like @racket[array-map], but requires at least one array argument and never @tech{broadcasts}.
|
||
|
||
TODO: more
|
||
}
|
||
|
||
@deftogether[(@defproc[(matrix* [M (Matrix Number)] [N (Matrix Number)] ...) (Matrix Number)]
|
||
@defproc[(matrix+ [M (Matrix Number)] [N (Matrix Number)] ...) (Matrix Number)]
|
||
@defproc[(matrix- [M (Matrix Number)] [N (Matrix Number)] ...) (Matrix Number)])]{
|
||
}
|
||
|
||
@defproc[(matrix-expt [M (Matrix Number)] [n Integer]) (Matrix Number)]{
|
||
Computes @racket[(matrix* M ...)] with @racket[n] arguments, but more efficiently.
|
||
@racket[M] must be a @racket[square-matrix?] and @racket[n] must be nonnegative.
|
||
}
|
||
|
||
@defproc[(matrix-scale [M (Matrix Number)] [x Number]) (Matrix Number)]{
|
||
}
|
||
|
||
@defproc[(matrix-sum [Ms (Listof (Matrix Number))]) (Matrix Number)]{
|
||
Like @racket[(apply matrix+ Ms)], but raises a runtime error when @racket[Ms] is empty.
|
||
}
|
||
|
||
@defproc[(matrix= [M0 (Matrix Number)] [M1 (Matrix Number)] [N (Matrix Number)] ...) Boolean]{
|
||
Returns @racket[#t] when its arguments are the same size and are equal entrywise.
|
||
|
||
See @racket[matrix-relative-error] and @racket[matrix-absolute-error] for equality testing that is
|
||
tolerant to floating-point error.
|
||
}
|
||
|
||
|
||
@;{==================================================================================================}
|
||
|
||
|
||
@section[#:tag "matrix:poly"]{Polymorphic Operations}
|
||
|
||
@defthing[matrix-ref Procedure]{
|
||
@;{(: matrix-ref (All (A) (Matrix A) Integer Integer -> A))}
|
||
}
|
||
|
||
@defthing[submatrix Procedure]{
|
||
@;{
|
||
(: submatrix (All (A) (Matrix A) Slice-Spec Slice-Spec -> (Matrix A)))
|
||
(define (submatrix a row-range col-range)
|
||
(array-slice-ref (ensure-matrix 'submatrix a) (list row-range col-range)))
|
||
}
|
||
}
|
||
|
||
@deftogether[(@defthing[matrix-row Procedure]
|
||
@defthing[matrix-col Procedure])]{
|
||
@;{
|
||
(: matrix-row (All (A) (Matrix A) Integer -> (Matrix A)))
|
||
(: matrix-col (All (A) (Matrix A) Integer -> (Matrix A)))
|
||
}
|
||
}
|
||
|
||
@defthing[matrix-diagonal Procedure]{
|
||
@;{(: matrix-diagonal (All (A) ((Matrix A) -> (Array A))))}
|
||
}
|
||
|
||
@deftogether[(@defthing[matrix-upper-triangle Procedure]
|
||
@defthing[matrix-lower-triangle Procedure])]{
|
||
@;{
|
||
(: matrix-upper-triangle (All (A) ((Matrix A) -> (Matrix (U A 0)))))
|
||
(: matrix-lower-triangle (All (A) ((Matrix A) -> (Matrix (U A 0)))))
|
||
}
|
||
}
|
||
|
||
@deftogether[(@defthing[matrix-rows Procedure]
|
||
@defthing[matrix-cols Procedure])]{
|
||
@;{
|
||
(: matrix-rows (All (A) (Matrix A) -> (Listof (Matrix A))))
|
||
(: matrix-cols (All (A) (Matrix A) -> (Listof (Matrix A))))
|
||
}
|
||
}
|
||
|
||
@deftogether[(@defthing[matrix-augment Procedure]
|
||
@defthing[matrix-stack Procedure])]{
|
||
@;{
|
||
(: matrix-augment (All (A) (Listof (Matrix A)) -> (Matrix A)))
|
||
(: matrix-stack (All (A) (Listof (Matrix A)) -> (Matrix A)))
|
||
}
|
||
}
|
||
|
||
@deftogether[(@defthing[matrix-map-rows Procedure]
|
||
@defthing[matrix-map-cols Procedure])]{
|
||
@;{
|
||
(: matrix-map-rows
|
||
(All (A B F) (case-> (((Matrix A) -> (Matrix B)) (Matrix A) -> (Matrix B))
|
||
(((Matrix A) -> (U #f (Matrix B))) (Matrix A) (-> F)
|
||
-> (U F (Matrix B))))))
|
||
|
||
(: matrix-map-cols
|
||
(All (A B F) (case-> (((Matrix A) -> (Matrix B)) (Matrix A) -> (Matrix B))
|
||
(((Matrix A) -> (U #f (Matrix B))) (Matrix A) (-> F)
|
||
-> (U F (Matrix B))))))
|
||
}
|
||
}
|
||
|
||
|
||
@;{==================================================================================================}
|
||
|
||
|
||
@section[#:tag "matrix:basic"]{Basic Operations}
|
||
|
||
@defthing[matrix-conjugate Procedure]{
|
||
@;{(: matrix-conjugate ((Matrix Number) -> (Matrix Number)))}
|
||
}
|
||
|
||
@deftogether[(@defthing[matrix-transpose Procedure]
|
||
@defthing[matrix-hermitian Procedure])]{
|
||
@;{
|
||
(: matrix-transpose (All (A) (Matrix A) -> (Matrix A)))
|
||
(: matrix-hermitian ((Matrix Number) -> (Matrix Number)))
|
||
}
|
||
}
|
||
|
||
@defthing[matrix-trace Procedure]{
|
||
@;{(: matrix-trace ((Matrix Number) -> Number))}
|
||
}
|
||
|
||
|
||
@;{==================================================================================================}
|
||
|
||
|
||
@section[#:tag "matrix:inner"]{Inner Product Space Operations}
|
||
|
||
TODO: explain: operations on the inner product space of matrices
|
||
|
||
TODO: explain: probably most useful to use these functions on row and column matrices
|
||
|
||
See @secref{matrix:op-norm} for similar functions (e.g. norms and angles) defined by considering
|
||
matrices as operators between inner product spaces consisting of column matrices.
|
||
|
||
@deftogether[(@defthing[matrix-1norm Procedure]
|
||
@defthing[matrix-2norm Procedure]
|
||
@defthing[matrix-inf-norm Procedure]
|
||
@defthing[matrix-norm Procedure])]{
|
||
@;{
|
||
(: matrix-1norm ((Matrix Number) -> Nonnegative-Real))
|
||
;; Manhattan, taxicab, or sum norm
|
||
|
||
(: matrix-2norm ((Matrix Number) -> Nonnegative-Real))
|
||
;; Frobenius, or Euclidean norm
|
||
|
||
(: matrix-inf-norm ((Matrix Number) -> Nonnegative-Real))
|
||
;; Maximum, or infinity norm
|
||
|
||
(: matrix-norm (case-> ((Matrix Number) -> Nonnegative-Real)
|
||
((Matrix Number) Real -> Nonnegative-Real)))
|
||
;; Any p norm
|
||
}
|
||
}
|
||
|
||
@defthing[matrix-dot Procedure]{
|
||
@;{
|
||
(: matrix-dot (case-> ((Matrix Number) -> Nonnegative-Real)
|
||
((Matrix Number) (Matrix Number) -> Number)))
|
||
;; Computes the Frobenius inner product of a matrix with itself or of two matrices
|
||
}
|
||
}
|
||
|
||
@defthing[matrix-cos-angle Procedure]{
|
||
@;{
|
||
(: matrix-cos-angle ((Matrix Number) (Matrix Number) -> Number))
|
||
;; Returns the cosine of the angle between two matrices w.r.t. the inner produce space induced by
|
||
;; the Frobenius inner product
|
||
}
|
||
}
|
||
|
||
@defthing[matrix-angle Procedure]{
|
||
Equivalent to @racket[(acos (matrix-cos-angle M0 M1))].
|
||
}
|
||
|
||
@defthing[matrix-normalize Procedure]{
|
||
@;{
|
||
(: matrix-normalize
|
||
(All (A) (case-> ((Matrix Number) -> (Matrix Number))
|
||
((Matrix Number) Real -> (Matrix Number))
|
||
((Matrix Number) Real (-> A) -> (U A (Matrix Number))))))
|
||
}
|
||
}
|
||
|
||
@deftogether[(@defthing[matrix-normalize-rows Procedure]
|
||
@defthing[matrix-normalize-cols Procedure])]{
|
||
@;{
|
||
(: matrix-normalize-rows
|
||
(All (A) (case-> ((Matrix Number) -> (Matrix Number))
|
||
((Matrix Number) Real -> (Matrix Number))
|
||
((Matrix Number) Real (-> A) -> (U A (Matrix Number))))))
|
||
|
||
(: matrix-normalize-cols
|
||
(All (A) (case-> ((Matrix Number) -> (Matrix Number))
|
||
((Matrix Number) Real -> (Matrix Number))
|
||
((Matrix Number) Real (-> A) -> (U A (Matrix Number))))))
|
||
}
|
||
}
|
||
|
||
@deftogether[(@defthing[matrix-rows-orthogonal? Procedure]
|
||
@defthing[matrix-cols-orthogonal? Procedure])]{
|
||
@;{
|
||
(: matrix-rows-orthogonal? ((Matrix Number) [Real (* 10 epsilon.0)] -> Boolean))
|
||
(: matrix-cols-orthogonal? ((Matrix Number) [Real (* 10 epsilon.0)] -> Boolean))
|
||
}
|
||
}
|
||
|
||
|
||
@;{==================================================================================================}
|
||
|
||
|
||
@section[#:tag "matrix:solve"]{Solving Systems of Equations}
|
||
|
||
@defthing[matrix-solve Procedure]{}
|
||
|
||
@defthing[matrix-inverse Procedure]{}
|
||
|
||
@defthing[matrix-invertible? Procedure]{}
|
||
|
||
@defthing[matrix-determinant Procedure]{}
|
||
|
||
|
||
@;{==================================================================================================}
|
||
|
||
|
||
@section[#:tag "matrix:row-alg"]{Row-Based Algorithms}
|
||
|
||
@defthing[matrix-gauss-elim Procedure]{}
|
||
|
||
@defthing[matrix-row-echelon Procedure]{}
|
||
|
||
@defthing[matrix-lu Procedure]{}
|
||
|
||
|
||
@;{==================================================================================================}
|
||
|
||
|
||
@section[#:tag "matrix:ortho-alg"]{Orthogonal Algorithms}
|
||
|
||
@defthing[matrix-gram-schmidt Procedure]{}
|
||
|
||
@defthing[matrix-basis-extension Procedure]{}
|
||
|
||
@defthing[matrix-qr Procedure]{}
|
||
|
||
|
||
@;{==================================================================================================}
|
||
|
||
|
||
@section[#:tag "matrix:op-norm"]{Operator Norms and Comparing Matrices}
|
||
|
||
@defproc[(matrix-op-1norm [M (Matrix Number)]) Nonnegative-Real]{
|
||
TODO: describe
|
||
|
||
When M is a column matrix, @racket[(matrix-op-1norm M)] is equivalent to @racket[(matrix-1norm M)].
|
||
}
|
||
|
||
@defproc[(matrix-op-2norm [M (Matrix Number)]) Nonnegative-Real]{
|
||
TODO: describe (spectral norm)
|
||
|
||
When M is a column matrix, @racket[(matrix-op-2norm M)] is equivalent to @racket[(matrix-2norm M)].
|
||
}
|
||
|
||
@defproc[(matrix-op-inf-norm [M (Matrix Number)]) Nonnegative-Real]{
|
||
TODO: describe
|
||
|
||
When M is a column matrix, @racket[(matrix-op-inf-norm M)] is equivalent to
|
||
@racket[(matrix-inf-norm M)].
|
||
}
|
||
|
||
@defproc[(matrix-basis-cos-angle [M0 (Matrix Number)] [M1 (Matrix Number)]) Number]{
|
||
Returns the cosine of the angle between the two subspaces spanned by @racket[M0] and @racket[M1].
|
||
|
||
When @racket[M0] and @racket[M1] are column matrices, @racket[(matrix-basis-cos-angle M0 M1)] is
|
||
equivalent to @racket[(matrix-cos-angle M0 M1)].
|
||
}
|
||
|
||
@defproc[(matrix-basis-angle [M0 (Matrix Number)] [M1 (Matrix Number)]) Number]{
|
||
Equivalent to @racket[(acos (matrix-basis-cos-angle M0 M1))].
|
||
}
|
||
|
||
@defparam[matrix-error-norm norm ((Matrix Number) -> Nonnegative-Real)]{
|
||
The norm used by @racket[matrix-relative-error] and @racket[matrix-absolute-error].
|
||
|
||
Besides being a true norm, @racket[norm] should also be @deftech{submultiplicative}:
|
||
@racketblock[(norm (matrix* M0 M1)) <= (* (norm A) (norm B))]
|
||
This additional triangle-like inequality makes it possible to prove error bounds for formulas that
|
||
involve matrix multiplication.
|
||
|
||
All operator norms (@racket[matrix-op-1norm], @racket[matrix-op-2norm], @racket[matrix-op-inf-norm])
|
||
are submultiplicative by definition, as is the Frobenius norm (@racket[matrix-2norm]).
|
||
}
|
||
|
||
@defproc[(matrix-absolute-error [M (Matrix Number)]
|
||
[R (Matrix Number)]
|
||
[norm ((Matrix Number) -> Nonnegative-Real) (matrix-error-norm)])
|
||
Nonnegative-Real]{
|
||
Basically equivalent to @racket[(norm (matrix- M R))], but handles non-rational flonums like
|
||
@racket[+inf.0] and @racket[+nan.0] specially.
|
||
|
||
See @racket[absolute-error] for the scalar version of this function.
|
||
}
|
||
|
||
@defproc[(matrix-relative-error [M (Matrix Number)]
|
||
[R (Matrix Number)]
|
||
[norm ((Matrix Number) -> Nonnegative-Real) (matrix-error-norm)])
|
||
Nonnegative-Real]{
|
||
Measures the error in @racket[M] relative to the true matrix @racket[R], under the norm @racket[norm].
|
||
Basically equivalent to @racket[(/ (norm (matrix- M R)) (norm R))], but handles non-rational flonums
|
||
like @racket[+inf.0] and @racket[+nan.0] specially, as well as the case @racket[(norm R) = 0].
|
||
|
||
See @racket[relative-error] for the scalar version of this function.
|
||
}
|
||
|
||
@defproc[(matrix-zero? [M (Matrix Number)] [eps Real (* 10 epsilon.0)]) Boolean]{
|
||
Returns @racket[#t] when @racket[M] is very close to a zero matrix (by default, within a few
|
||
epsilons). Equivalent to
|
||
@racketblock[(<= (matrix-absolute-error M (make-matrix m n 0)) eps)]
|
||
where @racket[m n] is the shape of @racket[M].
|
||
}
|
||
|
||
@defproc[(matrix-identity? [M (Matrix Number)] [eps Real (* 10 epsilon.0)]) Boolean]{
|
||
Returns @racket[#t] when @racket[M] is very close to the identity matrix (by default, within a few
|
||
epsilons).
|
||
Equivalent to
|
||
@racketblock[(and (square-matrix? M)
|
||
(<= (matrix-relative-error M (identity-matrix (square-matrix-size M)))
|
||
eps))]
|
||
}
|
||
|
||
@defproc[(matrix-orthonormal? [M (Matrix Number)] [eps Real (* 10 epsilon.0)]) Boolean]{
|
||
Returns @racket[#t] when @racket[M] is very close to being orthonormal; that is, when
|
||
@racket[(matrix* M (matrix-hermitian M))] is very close to an identity matrix.
|
||
In fact, @racket[(matrix-orthonormal? M eps)] is equivalent to
|
||
@racketblock[(matrix-identity? (matrix* M (matrix-hermitian M)) eps)]
|
||
}
|
||
|
||
@(close-eval typed-eval)
|
||
@(close-eval untyped-eval)
|