Finished matrix documentation, attendant fixes; please merge to 5.3.2
* Narrowed type of `submatrix' to only sensible argument types
* `matrix-invertible?' now returns #f when given a non-square matrix
instead of raising an error
* Allowed `matrix-diagonal' to operate on non-square matrices
(cherry picked from commit 24561e25e4)
This commit is contained in:
Neil Toronto
Ryan Culpepper
parent
8dd7ed2004
commit
6708c385e6
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@ -70,7 +70,8 @@
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[else
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(unsafe-array-ref a ((inst vector Index) i j))]))
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(: submatrix (All (A) (Matrix A) Slice-Spec Slice-Spec -> (Matrix A)))
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(: submatrix (All (A) (Matrix A) (U Slice (Sequenceof Integer)) (U Slice (Sequenceof Integer))
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-> (Array A)))
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(define (submatrix a row-range col-range)
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(array-slice-ref (ensure-matrix 'submatrix a) (list row-range col-range)))
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@ -120,11 +121,11 @@
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(: matrix-diagonal (All (A) ((Matrix A) -> (Array A))))
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(define (matrix-diagonal a)
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(define m (square-matrix-size a))
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(define-values (m n) (matrix-shape a))
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(define proc (unsafe-array-proc a))
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(array-default-strict
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(unsafe-build-array
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((inst vector Index) m)
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((inst vector Index) (fxmin m n))
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(λ: ([js : Indexes])
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(define i (unsafe-vector-ref js 0))
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(proc ((inst vector Index) i i))))))
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@ -305,8 +306,11 @@
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(: matrix-trace (case-> ((Matrix Real) -> Real)
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((Matrix Number) -> Number)))
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(define (matrix-trace a)
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(parameterize ([array-strictness #f])
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(array-all-sum (matrix-diagonal a))))
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(cond [(square-matrix? a)
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(parameterize ([array-strictness #f])
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(array-all-sum (matrix-diagonal a)))]
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[else
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(raise-argument-error 'matrix-trace "square-matrix?" a)]))
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;; ===================================================================================================
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;; Row/column operations
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@ -69,7 +69,8 @@
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(: matrix-invertible? ((Matrix Number) -> Boolean))
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(define (matrix-invertible? M)
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(not (zero? (matrix-determinant M))))
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(and (square-matrix? M)
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(not (zero? (matrix-determinant M)))))
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(: matrix-inverse (All (A) (case-> ((Matrix Real) -> (Matrix Real))
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((Matrix Real) (-> A) -> (U A (Matrix Real)))
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@ -6,7 +6,7 @@
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racket/list
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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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ann inst : λ: define: make-predicate -> case-> Nothing
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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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@ -41,9 +41,16 @@ For now, if you need speed, use the @racketmodname[typed/racket] language.
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@defmodule[math/matrix]
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Documentation for this module is currently under construction.
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Intro topics: definitions, case-> types, non-strictness
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Like all of @racketmodname[math], @racketmodname[math/matrix] is a work in progress.
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Most of the basic algorithms are implemented, but some are still in planning.
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Possibly the most useful unimplemented algorithms are
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@itemlist[@item{LUP decomposition (currently, LU decomposition is implemented, in @racket[matrix-lu])}
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@item{@racket[matrix-solve] for upper-triangular matrices}
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@item{Singular value decomposition (SVD)}
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@item{Eigendecomposition}
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@item{Decomposition-based solvers}
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@item{Pseudoinverse, least-squares fitting}]
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All of these are planned for the next Racket release.
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@local-table-of-contents[]
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@ -51,6 +58,76 @@ Intro topics: definitions, case-> types, non-strictness
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@;{==================================================================================================}
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@section[#:tag "matrix:intro"]{Introduction}
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From the point of view of the functions in @racketmodname[math/matrix], a @deftech{matrix} is an
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@racket[Array] with two axes and at least one entry, or an array for which @racket[matrix?] returns
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@racket[#t].
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Technically, a matrix's entries may be any type, and some fully polymorphic matrix functions such as
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@racket[matrix-row] and @racket[matrix-map] operate on any kind of matrix.
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Other functions, such as @racket[matrix+], require their input matrices to contain either
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@racket[Real] or @racket[Number] values.
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@subsection[#:tag "matrix:function-types"]{Function Types}
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The documentation for @racketmodname[math/matrix] functions use the type @racket[Matrix], a synonym
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of @racket[Array], when the function either requires that an argument is a @tech{matrix} or ensures
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that a return value is a matrix.
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Most functions that implement matrix algorithms are documented as accepting @racket[(Matrix Number)]
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values. This includes @racket[(Matrix Real)], which is a subtype. Most of these functions have a more
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precise type than is documented. For example, @racket[matrix-conjugate] actually has the type
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@racketblock[(case-> ((Matrix Real) -> (Matrix Real))
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((Matrix Number) -> (Matrix Number)))]
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even though it is documented as having the less precise type
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@racket[((Matrix Number) -> (Matrix Number))].
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Precise function types allow Typed Racket to prove more facts about @racketmodname[math/matrix]
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client programs. In particular, it is usually easy for it to prove that matrix expressions on real
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matrices return real matrices:
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@interaction[#:eval typed-eval
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(matrix-conjugate (matrix [[1 2 3] [4 5 6]]))]
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@subsection[#:tag "matrix:failure"]{Failure Arguments}
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In many matrix operations, such as inversion, failure is easy to detect during computation, but
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is just as expensive to detect ahead of time as the operation itself.
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In these cases, the functions implementing the operations accept an optional @deftech{failure thunk},
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or a zero-argument function that returns the result of the operation in case of failure.
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For example, the (simplified) type of @racket[matrix-inverse] is
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@racketblock[(All (F) (case-> ((Matrix Number) -> (Matrix Number))
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((Matrix Number) (-> F) -> (U F (Matrix Number)))))]
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Thus, if a failure thunk is given, the call site is required to check for return values of type
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@racket[F] explicitly.
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The default failure thunk, which raises an error, has type @racket[(-> Nothing)].
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For such failure thunks, @racket[(U F (Matrix Number))] is equivalent to @racket[(Matrix Number)],
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because @racket[Nothing] is part of every type. (In Racket, any expression may raise an error.)
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Thus, in this case, no explicit test for values of type @racket[F] is necessary (though of course they
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may be caught using @racket[with-handlers] or similar).
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@subsection[#:tag "matrix:broadcasting"]{Broadcasting}
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Pointwise matrix operations do not @tech{broadcast} their arguments when given matrices with
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different sizes:
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@interaction[#:eval typed-eval
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(matrix+ (identity-matrix 2) (matrix [[10]]))]
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If you need broadcasting, use array operations:
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@interaction[#:eval typed-eval
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(array+ (identity-matrix 2) (matrix [[10]]))]
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@subsection{Strictness}
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Functions exported by @racketmodname[math/matrix] return @tech{strict} or @tech{nonstrict} arrays
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based on the value of the @racket[array-strictness] parameter.
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See @secref{array:nonstrict} for details.
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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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@ -59,7 +136,7 @@ Equivalent to @racket[(Array A)], but used for values @racket[M] for which @rack
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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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Returns @racket[#t] when @racket[arr] is a @tech{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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@ -161,7 +238,9 @@ Returns a matrix with @racket[xs] along the diagonal and @racket[0] everywhere e
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The length of @racket[xs] must be positive.
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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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@define[block-diagonal-url]{http://en.wikipedia.org/wiki/Block_matrix#Block_diagonal_matrices}
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@margin-note*{@hyperlink[block-diagonal-url]{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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@ -172,7 +251,9 @@ The length of @racket[Xs] must be positive.
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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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@define[vandermonde-url]{http://en.wikipedia.org/wiki/Vandermonde_matrix}
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@margin-note*{@hyperlink[vandermonde-url]{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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@ -313,11 +394,11 @@ where each entry in @racket[M] is multiplied with @racket[z].
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(matrix-scale (matrix [[1 2] [3 4]]) 2)]
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}
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@defproc*[([(matrix-map [f (A -> R)] [arr0 (Matrix A)]) (Matrix R)]
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[(matrix-map [f (A B Ts ... -> R)] [arr0 (Matrix A)] [arr1 (Matrix B)] [arrs (Matrix Ts)]
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@defproc*[([(matrix-map [f (A -> R)] [M (Matrix A)]) (Matrix R)]
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[(matrix-map [f (A B Ts ... -> R)] [M0 (Matrix A)] [M1 (Matrix B)] [N (Matrix Ts)]
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...)
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(Matrix R)])]{
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Like @racket[array-map], but requires at least one array argument and never @tech{broadcasts}.
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Like @racket[array-map], but requires at least one matrix argument and never @tech{broadcasts}.
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@examples[#:eval untyped-eval
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(matrix-map sqr (matrix [[1 2] [3 4]]))
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(matrix-map + (matrix [[1 2] [3 4]])
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@ -344,47 +425,52 @@ tolerant to floating-point error.
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@defproc[(matrix-ref [M (Matrix A)] [i Integer] [j Integer]) A]{
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Returns the entry on row @racket[i] and column @racket[j].
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@examples[#:eval untyped-eval
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(define A (matrix ([1 2 3] [4 5 6])))
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(matrix-ref A 0 2)
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(matrix-ref A 1 2)]
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}
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@defthing[submatrix Procedure]{
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@;{ TODO
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(: submatrix (All (A) (Matrix A) Slice-Spec Slice-Spec -> (Matrix A)))
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(define (submatrix a row-range col-range)
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(array-slice-ref (ensure-matrix 'submatrix a) (list row-range col-range)))
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}
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TODO
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(define M (matrix ([1 2 3] [4 5 6])))
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(matrix-ref M 0 2)
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(matrix-ref M 1 2)]
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}
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@deftogether[(@defproc[(matrix-row [M (Matrix A)] [i Integer]) (Matrix A)]
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@defproc[(matrix-col [M (Matrix A)] [j Integer]) (Matrix A)])]{
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Returns the given row or column.
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Returns the @racket[i]th row or @racket[j]th column as a matrix.
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@examples[#:eval untyped-eval
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(define A (matrix ([1 2 3] [4 5 6])))
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(matrix-row A 1)
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(matrix-col A 0)]
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(define M (matrix ([1 2 3] [4 5 6])))
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(matrix-row M 1)
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(matrix-col M 0)]
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}
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@defproc[(submatrix [M (Matrix A)]
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[is (U Slice (Sequenceof Integer))]
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[js (U Slice (Sequenceof Integer))])
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(Array A)]{
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Returns a submatrix or subarray of @racket[M], where @racket[is] and @racket[js] specify
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respectively the rows and columns to keep. Like @racket[array-slice-ref], but constrained so the
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result has exactly two axes.
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@examples[#:eval typed-eval
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(submatrix (identity-matrix 5) (:: 1 #f 2) (::))
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(submatrix (identity-matrix 5) '() '(1 2 4))]
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Note that @racket[submatrix] may return an empty array, which is not a @tech{matrix}.
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}
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@defproc[(matrix-diagonal [M (Matrix A)]) (Array A)]{
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Returns array of the elements on the diagonal of the square matrix.
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Returns array of the entries on the diagonal of @racket[M].
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@examples[#:eval untyped-eval
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(define A (matrix ([1 2 3] [4 5 6] [7 8 9])))
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(matrix-diagonal A)]
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(matrix-diagonal
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(matrix ([1 2 3] [4 5 6] [7 8 9])))]
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}
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@deftogether[(@defproc[(matrix-upper-triangle [M (Matrix A)]) (Matrix A)]
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@defproc[(matrix-lower-triangle [M (Matrix A)]) (Matrix A)])]{
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@deftogether[(@defproc[(matrix-upper-triangle [M (Matrix A)]) (Matrix (U A 0))]
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@defproc[(matrix-lower-triangle [M (Matrix A)]) (Matrix (U A 0))])]{
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The function @racket[matrix-upper-triangle] returns an upper
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triangular matrix (entries below the diagonal are zero) with
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elements from the given matrix. Likewise the function
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@racket[matrix-lower-triangle] returns an lower triangular
|
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entries from the given matrix. Likewise the function
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@racket[matrix-lower-triangle] returns a lower triangular
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matrix.
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@examples[#:eval untyped-eval
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(define A (matrix ([1 2 3] [4 5 6] [7 8 9])))
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(matrix-upper-triangle A)
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(matrix-lower-triangle A)]
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(define M (array+ (array 1) (axis-index-array #(5 7) 1)))
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M
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(matrix-upper-triangle M)
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(matrix-lower-triangle M)]
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}
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@deftogether[(@defproc[(matrix-rows [M (Matrix A)]) (Listof (Matrix A))]
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@ -392,45 +478,61 @@ matrix.
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The functions respectively returns a list of the rows or columns
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of the matrix.
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@examples[#:eval untyped-eval
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(define A (matrix ([1 2 3] [4 5 6])))
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(matrix-rows A)
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(matrix-cols A)]
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(define M (matrix ([1 2 3] [4 5 6])))
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(matrix-rows M)
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(matrix-cols M)]
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}
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@deftogether[(@defproc[(matrix-augment [Ms (Listof (Matrix A))]) (Matrix A)]
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@defproc[(matrix-stack [Ms (Listof (Matrix A))]) (Matrix A)])]{
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The function @racket[matrix-augment] returns a matrix whose columns are
|
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the columns of the matrices in @racket[Ms]. This implies that the matrices
|
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the columns of the matrices in @racket[Ms]. The matrices
|
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in list must have the same number of rows.
|
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|
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The function @racket[matrix-stack] returns a matrix whose rows are
|
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the rows of the matrices in @racket[Ms]. This implies that the matrices
|
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the rows of the matrices in @racket[Ms]. The matrices
|
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in list must have the same number of columns.
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@examples[#:eval untyped-eval
|
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(define A (matrix ([1 1] [1 1])))
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(define B (matrix ([2 2] [2 2])))
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(define C (matrix ([3 3] [3 3])))
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(matrix-augment (list A B C))
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(matrix-stack (list A B C))]
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(define M0 (matrix ([1 1] [1 1])))
|
||||
(define M1 (matrix ([2 2] [2 2])))
|
||||
(define M2 (matrix ([3 3] [3 3])))
|
||||
(matrix-augment (list M0 M1 M2))
|
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(matrix-stack (list M0 M1 M2))]
|
||||
}
|
||||
|
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@deftogether[
|
||||
(@defproc[(matrix-map-rows
|
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[f ((Matrix A) -> (U #f (Matrix B)))] [M (Matrix A)] [fail (-> F) (λ () #f)]) (Matrix B)]
|
||||
@defproc[(matrix-map-cols
|
||||
[f ((Matrix A) -> (U #f (Matrix B)))] [M (Matrix A)] [fail (-> F) (λ () #f)]) (Matrix B)])]{
|
||||
In the simple case the function @racket[matrix-map-rows] applies the function @racket[f]
|
||||
to each row of @racket[M]. If the rows are called @racket[r0], @racket[r1], ... then
|
||||
the result matrix has the rows @racket[(f r0)], @racket[(f r1)], ... .
|
||||
In the three argument case, the result of @racket[(fail)] is used,
|
||||
if @racket[f] returns @racket[#f].
|
||||
|
||||
The function @racket[matrix-map-cols] works likewise but on rows.
|
||||
(@defproc*[([(matrix-map-rows
|
||||
[f ((Matrix A) -> (Matrix B))] [M (Matrix A)]) (Matrix B)]
|
||||
[(matrix-map-rows
|
||||
[f ((Matrix A) -> (U #f (Matrix B)))] [M (Matrix A)] [fail (-> F)])
|
||||
(U F (Matrix B))])])]{
|
||||
The two-argument case applies the function @racket[f] to each row of @racket[M].
|
||||
If the rows are called @racket[r0], @racket[r1], ..., the result matrix has the rows
|
||||
@racket[(f r0)], @racket[(f r1)], ....
|
||||
|
||||
@examples[#:eval untyped-eval
|
||||
(define A (matrix ([1 2 3] [4 5 6] [7 8 9] [10 11 12])))
|
||||
(define M (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)]
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||||
(matrix-map-rows double-row M)]
|
||||
|
||||
In the three argument case, if @racket[f] returns @racket[#f], the result of @racket[(fail)] is
|
||||
returned:
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||||
@interaction[#:eval typed-eval
|
||||
(define Z (make-matrix 4 4 0))
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||||
Z
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||||
(matrix-map-rows (λ: ([r : (Matrix Real)])
|
||||
(matrix-normalize r 2 (λ () #f)))
|
||||
Z
|
||||
(λ () 'FAILURE))]
|
||||
}
|
||||
|
||||
@deftogether[
|
||||
(@defproc*[([(matrix-map-cols
|
||||
[f ((Matrix A) -> (Matrix B))] [M (Matrix A)]) (Matrix B)]
|
||||
[(matrix-map-cols
|
||||
[f ((Matrix A) -> (U #f (Matrix B)))] [M (Matrix A)] [fail (-> F)])
|
||||
(U F (Matrix B))])])]{
|
||||
Like @racket[matrix-map-cols], but maps @racket[f] over columns.
|
||||
}
|
||||
|
||||
|
||||
|
|
@ -439,16 +541,16 @@ The function @racket[matrix-map-cols] works likewise but on rows.
|
|||
|
||||
@section[#:tag "matrix:basic"]{Basic Operations}
|
||||
|
||||
@defproc[(matrix-conjugate [M (Matrix A)]) (Matrix A)]{
|
||||
Returns a matrix where each element of the given matrix is conjugated.
|
||||
@defproc[(matrix-conjugate [M (Matrix Number)]) (Matrix Number)]{
|
||||
Returns a matrix where each entry of the given matrix is conjugated.
|
||||
@examples[#:eval untyped-eval
|
||||
(matrix-conjugate (matrix ([1 +i] [-1 2+i])))]
|
||||
}
|
||||
|
||||
@margin-note{@hyperlink["http://en.wikipedia.org/wiki/Transpose"]{Wikipedia: Transpose}}
|
||||
@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}}
|
||||
@defproc[(matrix-hermitian [M (Matrix Number)]) (Matrix Number)])]{
|
||||
@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.
|
||||
|
|
@ -457,71 +559,68 @@ For a real matrix these operations return the the same result.
|
|||
(matrix-hermitian (matrix ([1 +i] [2 +2i] [3 +3i])))]
|
||||
}
|
||||
|
||||
@margin-note{@hyperlink["http://en.wikipedia.org/wiki/Trace_(linear_algebra)"]{Wikipedia: Trace}}
|
||||
@defproc[(matrix-trace [M (Matrix Number)]) (Matrix Number)]{
|
||||
@margin-note*{@hyperlink["http://en.wikipedia.org/wiki/Trace_(linear_algebra)"]{Wikipedia: Trace}}
|
||||
@defproc[(matrix-trace [M (Matrix Number)]) Number]{
|
||||
Returns the trace of the square matrix. The trace of matrix is the
|
||||
the sum of the diagonal elements.
|
||||
the sum of the diagonal entries.
|
||||
@examples[#:eval untyped-eval
|
||||
(matrix-trace (matrix ([1 2] [3 4])))]
|
||||
}
|
||||
|
||||
|
||||
|
||||
@;{==================================================================================================}
|
||||
|
||||
|
||||
@section[#:tag "matrix:inner"]{Inner Product Space Operations}
|
||||
|
||||
The set of matrices of a given size forms a vector space.
|
||||
Since the vector space of @racket[mx1] matrices is isomorphic to
|
||||
the vector space of vectors of size @racket[m], any inner
|
||||
product (or norm) induce an inner product (or norm) on vectors.
|
||||
|
||||
Put differently, the following innner products and norm, even
|
||||
though defined on general matrices, work on vectors in the
|
||||
form of column and row matrices.
|
||||
The following functions treat matrices as vectors in an inner product space.
|
||||
It often makes most sense to use these vector-space functions only for
|
||||
@tech[#:key "row matrix"]{row matrices} and @tech[#:key "column matrix"]{column matrices}, which are
|
||||
essentially vectors as we normally think of them.
|
||||
There are exceptions, however, such as the fact that the Frobenius or Euclidean norm (implemented by
|
||||
@racket[matrix-2norm]) can be used to measure error between matrices in a way that meets certain
|
||||
reasonable criteria (specifically, it is submultiplicative).
|
||||
|
||||
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.
|
||||
|
||||
@margin-note{@hyperlink["http://en.wikipedia.org/wiki/Norm_(mathematics)"]{Wikipedia: Norm}}
|
||||
@deftogether[(@defproc[(matrix-1norm [M (Matrix Number)]) Number]
|
||||
@defproc[(matrix-2norm [M (Matrix Number)]) Number]
|
||||
@defproc[(matrix-inf-norm [M (Matrix Number)]) Number]
|
||||
@defproc*[([(matrix-norm [M (Matrix Number)]) Number]
|
||||
[(matrix-norm [M (Matrix Number)] [p Real]) Number])])]{
|
||||
The first three functions compute the L1-norm, the L2-norm, and, the L∞-norm respectively.
|
||||
@margin-note*{@hyperlink["http://en.wikipedia.org/wiki/Norm_(mathematics)"]{Wikipedia: Norm}}
|
||||
@deftogether[(@defproc[(matrix-1norm [M (Matrix Number)]) Nonnegative-Real]
|
||||
@defproc[(matrix-2norm [M (Matrix Number)]) Nonnegative-Real]
|
||||
@defproc[(matrix-inf-norm [M (Matrix Number)]) Nonnegative-Real]
|
||||
@defproc[(matrix-norm [M (Matrix Number)] [p Real 2]) Nonnegative-Real])]{
|
||||
Respectively compute the L@subscript{1} norm, L@subscript{2} norm, L@subscript{∞}, and
|
||||
L@subscript{p} norm.
|
||||
|
||||
The L1-norm is also known under the names Manhattan- or taxicab-norm.
|
||||
The L1-norm of a matrix is the sum of magnitudes of the entries in the matrix.
|
||||
The L@subscript{1} norm is also known under the names Manhattan or taxicab norm.
|
||||
The L@subscript{1} norm of a matrix is the sum of magnitudes of the entries in the matrix.
|
||||
|
||||
The L2-norm is also known under the names Euclidean- or Frobenius-norm.
|
||||
The L2-norm of a matrix is the square root of the sum of squares of
|
||||
The L@subscript{2} norm is also known under the names Euclidean or Frobenius norm.
|
||||
The L@subscript{2} norm of a matrix is the square root of the sum of squares of
|
||||
magnitudes of the entries in the matrix.
|
||||
|
||||
The L∞-norm is also known as the maximum- or infinity-norm.
|
||||
The L∞-norm computes the maximum magnitude of the entries in the matrix.
|
||||
The L@subscript{∞} norm is also known as the maximum or infinity norm.
|
||||
The L@subscript{∞} norm computes the maximum magnitude of the entries in the matrix.
|
||||
|
||||
The function @racket[matrix-norm] computes the Lp-norm.
|
||||
For a number @racket[p>=1] the @racket[p]th root of the sum
|
||||
of all entries to the @racket[p]th power.
|
||||
For @racket[p >= 1], @racket[matrix-norm] computes the L@subscript{@racket[p]} norm:
|
||||
the @racket[p]th root of the sum of all entry magnitudes to the @racket[p]th power.
|
||||
@;{MathJax would be nice to have in Scribble...}
|
||||
If no @racket[p] is given, the 2-norm (Eucledian) is used.
|
||||
@examples[#:eval untyped-eval
|
||||
(matrix-1norm (col-matrix [1 2]))
|
||||
(matrix-2norm (col-matrix [1 2]))
|
||||
(matrix-inf-norm (col-matrix [1 2]))
|
||||
(matrix-norm (col-matrix [1 2]) 3)]
|
||||
(matrix-norm (col-matrix [1 2]) 3)
|
||||
(matrix-norm (col-matrix [1 2]) +inf.0)]
|
||||
}
|
||||
|
||||
@defproc*[([(matrix-dot [M (Matrix Number)]) Nonnegative-Real]
|
||||
[(matrix-dot [M1 (Matrix Number)] [M2 (Matrix Number)]) Number])]{
|
||||
[(matrix-dot [M (Matrix Number)] [N (Matrix Number)]) Number])]{
|
||||
|
||||
The call @racket[(matrix-dot M1 M2)] computes the Frobenius inner product of the
|
||||
The call @racket[(matrix-dot M N)] computes the Frobenius inner product of the
|
||||
two matrices with the same shape.
|
||||
In other words the sum of @racket[(* a (conjugate b))] is computed where
|
||||
@racket[a] runs over the entries in @racket[M1] and @racket[b] runs over
|
||||
the corresponding entries in @racket[M2].
|
||||
@racket[a] runs over the entries in @racket[M] and @racket[b] runs over
|
||||
the corresponding entries in @racket[N].
|
||||
|
||||
The call @racket[(matrix-dot M)] computes @racket[(matrix-dot M M)] efficiently.
|
||||
@examples[#:eval untyped-eval
|
||||
|
|
@ -529,79 +628,130 @@ The call @racket[(matrix-dot M)] computes @racket[(matrix-dot M M)] efficiently.
|
|||
(+ (* 1 3) (* 2 4))]
|
||||
}
|
||||
|
||||
@defproc[(matrix-cos-angle [M (Matrix Number)]) Number]{
|
||||
@defproc[(matrix-cos-angle [M (Matrix Number)] [N (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. That is it returns
|
||||
|
||||
@racket[(/ (matrix-dot M N) (* (matrix-2norm M) (matrix-2norm N)))]
|
||||
|
||||
@examples[#:eval untyped-eval
|
||||
(define e1 (col-matrix [1 0]))
|
||||
(define e2 (col-matrix [0 1]))
|
||||
(matrix-cos-angle e1 e2)
|
||||
(matrix-cos-angle e1 (matrix+ e1 e2))]
|
||||
(define M (col-matrix [1 0]))
|
||||
(define N (col-matrix [0 1]))
|
||||
(matrix-cos-angle M N)
|
||||
(matrix-cos-angle M (matrix+ M N))]
|
||||
}
|
||||
|
||||
@defproc[(matrix-angle [M1 (Matrix Number)] [M2 (Matrix Number)]) Number]{
|
||||
Equivalent to @racket[(acos (matrix-cos-angle M0 M1))].
|
||||
@defproc[(matrix-angle [M (Matrix Number)] [N (Matrix Number)]) Number]{
|
||||
Equivalent to @racket[(acos (matrix-cos-angle M N))].
|
||||
@examples[#:eval untyped-eval
|
||||
(require racket/math) ; for radians->degrees
|
||||
(define e1 (col-matrix [1 0]))
|
||||
(define e2 (col-matrix [0 1]))
|
||||
(radians->degrees (matrix-angle e1 e2))
|
||||
(radians->degrees (matrix-angle e1 (matrix+ e1 e2)))]
|
||||
(require (only-in math/base radians->degrees))
|
||||
(define M (col-matrix [1 0]))
|
||||
(define N (col-matrix [0 1]))
|
||||
(radians->degrees (matrix-angle M N))
|
||||
(radians->degrees (matrix-angle M (matrix+ M N)))]
|
||||
}
|
||||
|
||||
@defproc[(matrix-normalize [M (Matrix Number)] [p Real 2] [fail (-> A) raise-argument-error]) (U (Matrix Number) A)]{
|
||||
To normalize a matrix is to scale it, such that the result has norm 1.
|
||||
|
||||
The call @racket[(matrix-normalize M p fail)] normalizes @racket[M] with respect to
|
||||
the @racket[Lp]-norm. If the matrix @racket[M] has norm 0, the result of calling
|
||||
the thunk @racket[fail] is returned.
|
||||
|
||||
If no fail-thunk is given, an argument-error exception is raised.
|
||||
If no @racket[p] the L2-norm (Euclidean norm) is used.
|
||||
@examples[#:eval untyped-eval
|
||||
(require racket/math) ; for radians->degrees
|
||||
@defproc[(matrix-normalize [M (Matrix Number)] [p Real 2] [fail (-> F) (λ () (error ...))])
|
||||
(U F (Matrix Number))]{
|
||||
Normalizes @racket[M] with respect to the L@subscript{@racket[p]} norm.
|
||||
@examples[#:eval typed-eval
|
||||
(matrix-normalize (col-matrix [1 1]))
|
||||
(matrix-normalize (col-matrix [1 1]) 1)
|
||||
(matrix-normalize (col-matrix [1 1]) +inf.0)]
|
||||
The result of applying the @tech{failure thunk} @racket[fail] is returned if @racket[M]'s norm is
|
||||
zero.
|
||||
}
|
||||
|
||||
@deftogether[(@defproc[(matrix-normalize-rows [M (matrix Number)] [p Real 2] [fail (-> A) raise-argument-error]) (Matrix Number)]
|
||||
@defproc[(matrix-normalize-cols [M (matrix Number)] [p Real 2] [fail (-> A) raise-argument-error]) (Matrix Number)])]{
|
||||
@deftogether[(@defproc[(matrix-normalize-rows [M (Matrix Number)]
|
||||
[p Real 2]
|
||||
[fail (-> F) (λ () (error ...))]) (Matrix Number)]
|
||||
@defproc[(matrix-normalize-cols [M (Matrix Number)]
|
||||
[p Real 2]
|
||||
[fail (-> F) (λ () (error ...))]) (Matrix Number)])]{
|
||||
As @racket[matrix-normalize] but each row or column is normalized separately.
|
||||
The result is this a matrix with unit vectors as rows or columns.
|
||||
@examples[#:eval untyped-eval
|
||||
(require racket/math) ; for radians->degrees
|
||||
The result is a matrix with unit vectors as rows or columns.
|
||||
@examples[#:eval typed-eval
|
||||
(matrix-normalize-rows (matrix [[1 2] [2 4]]))
|
||||
(matrix-normalize-cols (matrix [[1 2] [2 4]]))]
|
||||
The result of applying the @tech{failure thunk} @racket[fail] is returned if the norm of any row
|
||||
or column in @racket[M] is zero.
|
||||
}
|
||||
|
||||
@deftogether[(@defproc[(matrix-rows-orthogonal? [M (Matrix Number)] [eps Real (* 10 epsilon.0)]) Boolean]
|
||||
@defproc[(matrix-cols-orthogonal? [M (Matrix Number)] [eps Real (* 10 epsilon.0)]) Boolean])]{
|
||||
|
||||
@deftogether[(@defproc[(matrix-rows-orthogonal? [M (Matrix Number)]
|
||||
[eps Real (* 10 epsilon.0)])
|
||||
Boolean]
|
||||
@defproc[(matrix-cols-orthogonal? [M (Matrix Number)]
|
||||
[eps Real (* 10 epsilon.0)])
|
||||
Boolean])]{
|
||||
Returns @racket[#t] if the rows or columns of @racket[M]
|
||||
are very close of being orthogonal (by default a few epsilons).
|
||||
@examples[#:eval untyped-eval
|
||||
(require racket/math) ; for radians->degrees
|
||||
(matrix-rows-orthogonal? (matrix [[1 1] [-1 1]]))
|
||||
(matrix-cols-orthogonal? (matrix [[1 1] [-1 1]]))]
|
||||
}
|
||||
|
||||
|
||||
|
||||
@;{==================================================================================================}
|
||||
|
||||
|
||||
@section[#:tag "matrix:solve"]{Solving Systems of Equations}
|
||||
|
||||
@defthing[matrix-solve Procedure]{}
|
||||
@defproc[(matrix-solve [M (Matrix Number)] [B (Matrix Number)] [fail (-> F) (λ () (error ...))])
|
||||
(U F (Matrix Number))]{
|
||||
Returns the matrix @racket[X] for which @racket[(matrix* M X)] is @racket[B].
|
||||
@racket[M] and @racket[B] must have the same number of rows.
|
||||
|
||||
@defthing[matrix-inverse Procedure]{}
|
||||
It is typical for @racket[B] (and thus @racket[X]) to be a @tech{column matrix}, but not required.
|
||||
If @racket[B] is not a column matrix, @racket[matrix-solve] solves for all the columns in @racket[B]
|
||||
simultaneously.
|
||||
|
||||
@defthing[matrix-invertible? Procedure]{}
|
||||
@examples[#:eval typed-eval
|
||||
(define M (matrix [[7 5] [3 -2]]))
|
||||
(define B0 (col-matrix [3 22]))
|
||||
(define B1 (col-matrix [19 4]))
|
||||
(matrix-solve M B0)
|
||||
(matrix* M (col-matrix [4 -5]))
|
||||
(matrix-solve M B1)
|
||||
(matrix-cols (matrix-solve M (matrix-augment (list B0 B1))))]
|
||||
|
||||
@defthing[matrix-determinant Procedure]{}
|
||||
@racket[matrix-solve] does not solve overconstrained or underconstrained systems, meaning that
|
||||
@racket[M] must be invertible.
|
||||
If @racket[M] is not invertible, the result of applying the @tech{failure thunk} @racket[fail] is
|
||||
returned.
|
||||
|
||||
@racket[matrix-solve] is implemented using @racket[matrix-gauss-elim] to preserve exactness in its
|
||||
output, with partial pivoting for greater numerical stability when @racket[M] is not exact.
|
||||
|
||||
See @racket[vandermonde-matrix] for an example that uses @racket[matrix-solve] to compute Legendre
|
||||
polynomials.
|
||||
}
|
||||
|
||||
@define[inverse-url]{http://en.wikipedia.org/wiki/Invertible_matrix}
|
||||
@margin-note*{@hyperlink[inverse-url]{Wikipedia: Invertible Matrix}}
|
||||
@defproc[(matrix-inverse [M (Matrix Number)] [fail (-> F) (λ () (error ...))]) (U F (Matrix Number))]{
|
||||
Returns the inverse of @racket[M] if it exists; otherwise returns the result of applying the
|
||||
@tech{failure thunk} @racket[fail].
|
||||
@examples[#:eval typed-eval
|
||||
(matrix-inverse (identity-matrix 3))
|
||||
(matrix-inverse (matrix [[7 5] [3 -2]]))
|
||||
(matrix-inverse (matrix [[1 2] [10 20]]))
|
||||
(matrix-inverse (matrix [[1 2] [10 20]]) (λ () #f))]
|
||||
}
|
||||
|
||||
@defproc[(matrix-invertible? [M (Matrix Number)]) Boolean]{
|
||||
Returns @racket[#t] when @racket[M] is a @racket[square-matrix?] and @racket[(matrix-determinant M)]
|
||||
is nonzero.
|
||||
}
|
||||
|
||||
@margin-note*{@hyperlink["http://en.wikipedia.org/wiki/Determinant"]{Wikipedia: Determinant}}
|
||||
@defproc[(matrix-determinant [M (Matrix Number)]) Number]{
|
||||
Returns the determinant of @racket[M], which must be a @racket[square-matrix?].
|
||||
@examples[#:eval typed-eval
|
||||
(matrix-determinant (diagonal-matrix '(1 2 3 4)))
|
||||
(* 1 2 3 4)
|
||||
(matrix-determinant (matrix [[1 2] [10 20]]))
|
||||
(matrix-determinant (col-matrix [1 2]))]
|
||||
}
|
||||
|
||||
|
||||
@;{==================================================================================================}
|
||||
|
|
@ -609,11 +759,87 @@ are very close of being orthogonal (by default a few epsilons).
|
|||
|
||||
@section[#:tag "matrix:row-alg"]{Row-Based Algorithms}
|
||||
|
||||
@defthing[matrix-gauss-elim Procedure]{}
|
||||
@define[gauss-url]{http://en.wikipedia.org/wiki/Gaussian_elimination}
|
||||
@define[gauss-jordan-url]{http://en.wikipedia.org/wiki/Gauss%E2%80%93Jordan_elimination}
|
||||
|
||||
@defthing[matrix-row-echelon Procedure]{}
|
||||
@margin-note*{@hyperlink[gauss-url]{Wikipedia: Gaussian elimination}}
|
||||
@defproc[(matrix-gauss-elim [M (Matrix Number)]
|
||||
[jordan? Any #f]
|
||||
[unitize-pivot? Any #f]
|
||||
[pivoting (U 'first 'partial) 'partial])
|
||||
(Values (Matrix Number) (Listof Index))]{
|
||||
@margin-note*{@hyperlink[gauss-jordan-url]{Wikipedia: Gauss-Jordan elimination}}
|
||||
Implements Gaussian elimination or Gauss-Jordan elimination.
|
||||
|
||||
@defthing[matrix-lu Procedure]{}
|
||||
If @racket[jordan?] is true, row operations are done both above and below the pivot.
|
||||
If @racket[unitize-pivot?] is true, the pivot's row is scaled so that the pivot value is @racket[1].
|
||||
When both are true, the algorithm is called Gauss-Jordan elimination, and the result matrix is in
|
||||
@italic{reduced} row echelon form.
|
||||
|
||||
If @racket[pivoting] is @racket['first], the first nonzero entry in the current column is used as the
|
||||
pivot. If @racket[pivoting] is @racket['partial], the largest-magnitude nonzero entry is used, which
|
||||
improves numerical stability on average when @racket[M] contains inexact entries.
|
||||
|
||||
The first return value is the result of Gaussian elimination.
|
||||
|
||||
The second return value is a list of indexes of columns that did not have a nonzero pivot.
|
||||
|
||||
See @racket[matrix-row-echelon] for examples.
|
||||
}
|
||||
|
||||
@define[row-echelon-url]{http://en.wikipedia.org/wiki/Row_echelon_form}
|
||||
|
||||
@margin-note*{@hyperlink[row-echelon-url]{Wikipedia: Row echelon form}}
|
||||
@defproc[(matrix-row-echelon [M (Matrix Number)]
|
||||
[jordan? Any #f]
|
||||
[unitize-pivot? Any #f]
|
||||
[pivoting (U 'first 'partial) 'partial])
|
||||
(Matrix Number)]{
|
||||
Like @racket[matrix-gauss-elim], but returns only the result of Gaussian elimination.
|
||||
@examples[#:eval typed-eval
|
||||
(define M (matrix [[2 1 -1] [-3 -1 2] [-2 1 2]]))
|
||||
(matrix-row-echelon M)
|
||||
(matrix-row-echelon M #t)
|
||||
(matrix-identity? (matrix-row-echelon M #t #t))]
|
||||
The last example shows that @racket[M] is invertible.
|
||||
|
||||
Using @racket[matrix-row-echelon] to solve a system of linear equations (without checking for
|
||||
failure):
|
||||
@interaction[#:eval typed-eval
|
||||
(define B (col-matrix [8 -11 -3]))
|
||||
(define MB (matrix-augment (list M B)))
|
||||
(matrix-col (matrix-row-echelon MB #t #t) 3)
|
||||
(matrix-solve M B)]
|
||||
|
||||
Using @racket[matrix-row-echelon] to invert a matrix (also without checking for failure):
|
||||
@interaction[#:eval typed-eval
|
||||
(define MI (matrix-augment (list M (identity-matrix 3))))
|
||||
(submatrix (matrix-row-echelon MI #t #t) (::) (:: 3 #f))
|
||||
(matrix-inverse M)]
|
||||
}
|
||||
|
||||
@define[lu-url]{http://en.wikipedia.org/wiki/LU_decomposition}
|
||||
@margin-note*{@hyperlink[lu-url]{Wikipedia: LU decomposition}}
|
||||
@defproc[(matrix-lu [M (Matrix Number)] [fail (-> F) (λ () (error ...))])
|
||||
(Values (U F (Matrix Number)) (Matrix Number))]{
|
||||
Returns the LU decomposition of @racket[M] (which must be a @racket[square-matrix?]) if one exists.
|
||||
An LU decomposition exists if @racket[M] can be put in row-echelon form without swapping rows.
|
||||
|
||||
Because @racket[matrix-lu] returns a @italic{unit} lower-triangular matrix (i.e. a lower-triangular
|
||||
matrix with only ones on the diagonal), the decomposition is unique if it exists.
|
||||
|
||||
@examples[#:eval typed-eval
|
||||
(define-values (L U)
|
||||
(matrix-lu (matrix [[4 3] [6 3]])))
|
||||
(values L U)
|
||||
(matrix* L U)]
|
||||
|
||||
If @racket[M] does not have an LU decomposition, the first result is the result of applying the
|
||||
@tech{failure thunk} @racket[fail], and the second result is the original argument @racket[M]:
|
||||
@interaction[#:eval typed-eval
|
||||
(matrix-lu (matrix [[0 1] [1 1]]))
|
||||
(matrix-lu (matrix [[0 1] [1 1]]) (λ () #f))]
|
||||
}
|
||||
|
||||
|
||||
@;{==================================================================================================}
|
||||
|
|
@ -621,57 +847,143 @@ are very close of being orthogonal (by default a few epsilons).
|
|||
|
||||
@section[#:tag "matrix:ortho-alg"]{Orthogonal Algorithms}
|
||||
|
||||
@defthing[matrix-gram-schmidt Procedure]{}
|
||||
@define[gram-schmidt-url]{http://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process}
|
||||
@define[reortho-pdf]{http://www.cerfacs.fr/algor/reports/2002/TR_PA_02_33.pdf}
|
||||
|
||||
@defthing[matrix-basis-extension Procedure]{}
|
||||
@margin-note*{@hyperlink[gram-schmidt-url]{Wikipedia: Gram-Schmidt process}}
|
||||
@defproc[(matrix-gram-schmidt [M (Matrix Number)] [normalize? Any #f] [start-col Integer 0])
|
||||
(Array Number)]{
|
||||
Returns an array whose columns are orthogonal and span the same subspace as @racket[M]'s columns.
|
||||
The number of columns in the result is the rank of @racket[M].
|
||||
If @racket[normalize?] is true, the columns are also normalized.
|
||||
|
||||
@margin-note{@hyperlink["http://en.wikipedia.org/wiki/QR_decomposition"]{Wikipedia: QR decomposition}}
|
||||
@defproc*[([(matrix-qr [M (Matrix Real)]) (Values (Matrix Real) (Matrix Real))]
|
||||
[(matrix-qr [M (Matrix Real)] [full Any]) (Values (Matrix Real) (Matrix Real))]
|
||||
[(matrix-qr [M (Matrix Number)]) (Values (Matrix Number) (Matrix Number))]
|
||||
[(matrix-qr [M (Matrix Number)] [full Any]) (Values (Matrix Number) (Matrix Number))])]{
|
||||
@examples[#:eval typed-eval
|
||||
(define M
|
||||
(matrix [[12 -51 4]
|
||||
[ 6 167 -68]
|
||||
[-4 24 -41]]))
|
||||
(matrix-gram-schmidt M)
|
||||
(matrix-gram-schmidt M #t)
|
||||
(matrix-cols-orthogonal? (matrix-gram-schmidt M))
|
||||
(matrix-orthonormal? (matrix-gram-schmidt M #t))]
|
||||
|
||||
Examples with rank-deficient matrices:
|
||||
@interaction[#:eval typed-eval
|
||||
(matrix-gram-schmidt (matrix [[ 1 -2 1]
|
||||
[-2 4 9]
|
||||
[ 3 -6 7]]))
|
||||
(matrix-gram-schmidt (make-matrix 3 3 0))]
|
||||
|
||||
When @racket[start-col] is positive, the Gram-Schmidt process is begun on column @racket[start-col]
|
||||
(but still using the previous columns to orthogonalize the remaining columns).
|
||||
This feature is generally not directly useful, but is used in the implementation of
|
||||
@racket[matrix-basis-extension].
|
||||
|
||||
@margin-note*{* @bold{On the round-off error analysis of the Gram-Schmidt algorithm with
|
||||
reorthogonalization.},
|
||||
Luc Giraud, Julien Langou and Miroslav Rozloznik. 2002. @hyperlink[reortho-pdf]{(PDF)}}
|
||||
While Gram-Schmidt with inexact matrices is known to be unstable, using it twice tends to remove
|
||||
instabilities:*
|
||||
@interaction[#:eval typed-eval
|
||||
(define M (matrix [[0.70000 0.70711]
|
||||
[0.70001 0.70711]]))
|
||||
(matrix-orthonormal?
|
||||
(matrix-gram-schmidt M #t))
|
||||
(matrix-orthonormal?
|
||||
(matrix-gram-schmidt (matrix-gram-schmidt M) #t))]
|
||||
}
|
||||
|
||||
@defproc[(matrix-basis-extension [M (Matrix Number)]) (Array Number)]{
|
||||
Returns additional orthogonal columns which, if augmented with @racket[M], would result in an
|
||||
orthogonal matrix of full rank. If @racket[M]'s columns are normalized, the result's columns are
|
||||
normalized.
|
||||
}
|
||||
|
||||
@define[qr-url]{http://en.wikipedia.org/wiki/QR_decomposition}
|
||||
|
||||
@margin-note*{@hyperlink["http://en.wikipedia.org/wiki/QR_decomposition"]{Wikipedia: QR decomposition}}
|
||||
@defproc*[([(matrix-qr [M (Matrix Number)]) (Values (Matrix Number) (Matrix Number))]
|
||||
[(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
|
||||
the matrices @racket[Q] and @racket[R]. If @racket[full?] is @racket[#f], 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.}
|
||||
@margin-note*{An @italic{orthonormal} matrix has columns which are orthoginal, 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}
|
||||
See the Wikipedia entry on @hyperlink[qr-url]{QR decomposition}
|
||||
for more details.
|
||||
|
||||
@examples[#:eval typed-eval
|
||||
(define M
|
||||
(matrix [[12 -51 4]
|
||||
[ 6 167 -68]
|
||||
[-4 24 -41]]))
|
||||
(define-values (Q R) (matrix-qr M))
|
||||
(values Q R)
|
||||
(matrix= M (matrix* Q R))]
|
||||
|
||||
The difference between full and reduced decompositions:
|
||||
@interaction[#:eval typed-eval
|
||||
(define M
|
||||
(matrix [[12 -51]
|
||||
[ 6 167]
|
||||
[-4 24]]))
|
||||
(define-values (Q1 R1) (matrix-qr M #f))
|
||||
(define-values (Q2 R2) (matrix-qr M #t))
|
||||
(values Q1 R1)
|
||||
(values Q2 R2)
|
||||
(matrix= M (matrix* Q1 R1))
|
||||
(matrix= M (matrix* Q2 R2))]
|
||||
|
||||
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, and, Miroslav Rozloznik.
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
@;{==================================================================================================}
|
||||
|
||||
|
||||
@define[op-norm-url]{http://en.wikipedia.org/wiki/Matrix_norm#Induced_norm}
|
||||
|
||||
@section[#:tag "matrix:op-norm"]{Operator Norms and Comparing Matrices}
|
||||
|
||||
@secref{matrix:inner} describes functions that deal with matrices as vectors in
|
||||
an inner product space. This section describes functions that deal with matrices as
|
||||
@italic{linear operators}, or as functions from column matrices to column matrices.
|
||||
|
||||
@margin-note*{@hyperlink[op-norm-url]{Wikipedia: Induced norm}}
|
||||
In this setting, a norm is the largest relative change in magnitude an operator (i.e. matrix) can
|
||||
effect on a column matrix, where ``magnitude'' is defined by a vector norm.
|
||||
(See the Wikipedia article linked to in the margin for a formal definition.)
|
||||
Matrix norms that are defined in terms of a vector norm are called @italic{induced norms}, or
|
||||
@deftech{operator norms}.
|
||||
|
||||
@defproc[(matrix-op-1norm [M (Matrix Number)]) Nonnegative-Real]{
|
||||
TODO: describe
|
||||
The @tech{operator norm} induced by the vector norm @racket[matrix-1norm].
|
||||
|
||||
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)
|
||||
The @tech{operator norm} induced by the vector norm @racket[matrix-2norm].
|
||||
|
||||
This function is currently undefined because a required algorithm (singular value decomposition or
|
||||
eigendecomposition) is not yet implemented in @racketmodname[math/matrix].
|
||||
|
||||
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
|
||||
The @tech{operator norm} induced by the vector norm @racket[matrix-inf-norm].
|
||||
|
||||
When M is a column matrix, @racket[(matrix-op-inf-norm M)] is equivalent to
|
||||
@racket[(matrix-inf-norm M)].
|
||||
|
|
@ -680,16 +992,22 @@ When M is a column matrix, @racket[(matrix-op-inf-norm M)] is equivalent to
|
|||
@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].
|
||||
|
||||
This function is currently undefined because a required algorithm (singular value decomposition or
|
||||
eigendecomposition) is not yet implemented in @racketmodname[math/matrix].
|
||||
|
||||
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))].
|
||||
|
||||
The function is currently undefined because @racket[matrix-basis-cos-angle] is currently undefined.
|
||||
}
|
||||
|
||||
@defparam[matrix-error-norm norm ((Matrix Number) -> Nonnegative-Real)]{
|
||||
The norm used by @racket[matrix-relative-error] and @racket[matrix-absolute-error].
|
||||
The default value is @racket[matrix-op-inf-norm].
|
||||
|
||||
Besides being a true norm, @racket[norm] should also be @deftech{submultiplicative}:
|
||||
@racketblock[(norm (matrix* M0 M1)) <= (* (norm A) (norm B))]
|
||||
|
|
@ -740,7 +1058,7 @@ Equivalent to
|
|||
@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
|
||||
Equivalent to
|
||||
@racketblock[(matrix-identity? (matrix* M (matrix-hermitian M)) eps)]
|
||||
}
|
||||
|
||||
|
|
|
|||
|
|
@ -17,17 +17,18 @@ for working with numbers and collections of numbers. These include
|
|||
@item{@racketmodname[math/bigfloat]: Arbitrary-precision floating-point functions}
|
||||
@item{@racketmodname[math/number-theory]: Number-theoretic functions}
|
||||
@item{@racketmodname[math/array]: Functional arrays for operating on large rectangular data sets}
|
||||
@item{@racketmodname[math/matrix]: Linear algebra functions for arrays @bold{(currently being documented)}}
|
||||
@item{@racketmodname[math/matrix]: Linear algebra functions for arrays}
|
||||
@item{@racketmodname[math/distributions]: Probability distributions}
|
||||
@item{@racketmodname[math/statistics]: Statistical functions}
|
||||
]
|
||||
|
||||
With this library, we hope to support a wide variety of applied mathematics in
|
||||
Racket, including simulation, statistical inference, signal processing, and
|
||||
combinatorics. If you find it lacking for your variety of mathematics,
|
||||
please contact us or post to one of the
|
||||
@hyperlink["http://racket-lang.org/community.html"]{mailing lists}
|
||||
to make suggestions or submit patches.
|
||||
combinatorics. If you find it lacking for your variety of mathematics, please
|
||||
@itemlist[@item{Visit the @hyperlink["https://github.com/plt/racket/wiki/Math-Library-Features"]{Math Library Features wiki page}
|
||||
to see what is planned.}
|
||||
@item{Contact us or post to one of the @hyperlink["http://racket-lang.org/community.html"]{mailing lists} to
|
||||
make suggestions or submit patches.}]
|
||||
|
||||
@bold{This is a Typed Racket library.} It is most efficient to use it in Typed
|
||||
Racket, so that contracts are checked statically. However, almost all of it can
|
||||
|
|
|
|||
|
|
@ -96,7 +96,7 @@
|
|||
(check-true (array-strict? (matrix-col M i))))
|
||||
|
||||
(for*: ([M (list nonstrict-2x2-arr strict-2x2-arr)]
|
||||
[spec (list '(0) 0)])
|
||||
[spec (list '(0) (:: #f #f 2))])
|
||||
(check-true (array-strict? (submatrix M (::) spec))))
|
||||
)
|
||||
|
||||
|
|
@ -153,7 +153,7 @@
|
|||
(check-false (array-strict? R))))
|
||||
|
||||
(for*: ([M (list nonstrict-2x2-arr strict-2x2-arr)]
|
||||
[spec (list '(0) 0)])
|
||||
[spec (list '(0) (:: #f #f 2))])
|
||||
(check-false (array-strict? (submatrix M (::) spec))))
|
||||
|
||||
(for*: ([M (list nonstrict-2x2-arr strict-2x2-arr)]
|
||||
|
|
|
|||
Loading…
Reference in New Issue
Block a user