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)
138 lines
5.4 KiB
Racket
138 lines
5.4 KiB
Racket
#lang typed/racket/base
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Two of the functions defined here currently just raise an error: `matrix-op-2norm' and
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`matrix-op-angle'. They need to compute, respectively, the maximum and minimum singular values of
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their matrix argument.
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See "How to Measure Errors" in the LAPACK manual for more details:
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http://www.netlib.org/lapack/lug/node75.html
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http://www.netlib.org/lapack/lug/node76.html
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|#
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(require racket/list
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racket/fixnum
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math/flonum
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"matrix-types.rkt"
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"matrix-arithmetic.rkt"
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"matrix-constructors.rkt"
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"matrix-basic.rkt"
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"utils.rkt"
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"../array/array-struct.rkt"
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"../array/array-pointwise.rkt"
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"../array/array-fold.rkt"
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)
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(provide
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;; Operator norms
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matrix-op-1norm
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matrix-op-2norm
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matrix-op-inf-norm
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matrix-basis-cos-angle
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matrix-basis-angle
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;; Error measurement
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matrix-error-norm
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matrix-absolute-error
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matrix-relative-error
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;; Approximate predicates
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matrix-zero?
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matrix-identity?
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matrix-orthonormal?
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)
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(: matrix-op-1norm ((Matrix Number) -> Nonnegative-Real))
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;; When M is a column matrix, this is equivalent to matrix-1norm
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(define (matrix-op-1norm M)
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(assert (apply max (map matrix-1norm (matrix-cols M))) nonnegative?))
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(: matrix-op-2norm ((Matrix Number) -> Nonnegative-Real))
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;; When M is a column matrix, this is equivalent to matrix-2norm
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(define (matrix-op-2norm M)
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;(matrix-max-singular-value M)
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;(sqrt (matrix-max-eigenvalue M))
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(error 'unimplemented))
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(: matrix-op-inf-norm ((Matrix Number) -> Nonnegative-Real))
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;; When M is a column matrix, this is equivalent to matrix-inf-norm
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(define (matrix-op-inf-norm M)
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(assert (apply max (map matrix-1norm (matrix-rows M))) nonnegative?))
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(: matrix-basis-cos-angle (case-> ((Matrix Real) (Matrix Real) -> Real)
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((Matrix Number) (Matrix Number) -> Number)))
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;; Returns the angle between the two subspaces spanned by the two given sets of column vectors
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(define (matrix-basis-cos-angle M R)
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;(matrix-min-singular-value (matrix* (matrix-hermitian M) R))
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(error 'unimplemented))
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(: matrix-basis-angle (case-> ((Matrix Real) (Matrix Real) -> Real)
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((Matrix Number) (Matrix Number) -> Number)))
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;; Returns the angle between the two subspaces spanned by the two given sets of column vectors
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(define (matrix-basis-angle M R)
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(acos (matrix-basis-cos-angle M R)))
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;; ===================================================================================================
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;; Error measurement
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(: matrix-error-norm (Parameterof ((Matrix Number) -> Nonnegative-Real)))
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(define matrix-error-norm (make-parameter matrix-op-inf-norm))
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(: matrix-absolute-error
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(case-> ((Matrix Number) (Matrix Number) -> Nonnegative-Real)
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((Matrix Number) (Matrix Number) ((Matrix Number) -> Nonnegative-Real)
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-> Nonnegative-Real)))
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(define (matrix-absolute-error M R [norm (matrix-error-norm)])
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(define-values (m n) (matrix-shapes 'matrix-absolute-error M R))
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(array-strict! M)
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(array-strict! R)
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(cond [(array-all-and (inline-array-map eqv? M R)) 0]
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[(and (array-all-and (inline-array-map number-rational? M))
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(array-all-and (inline-array-map number-rational? R)))
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(norm (matrix- (inline-array-map inexact->exact M)
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(inline-array-map inexact->exact R)))]
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[else +inf.0]))
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(: matrix-relative-error
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(case-> ((Matrix Number) (Matrix Number) -> Nonnegative-Real)
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((Matrix Number) (Matrix Number) ((Matrix Number) -> Nonnegative-Real)
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-> Nonnegative-Real)))
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(define (matrix-relative-error M R [norm (matrix-error-norm)])
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(define-values (m n) (matrix-shapes 'matrix-relative-error M R))
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(array-strict! M)
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(array-strict! R)
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(cond [(array-all-and (inline-array-map eqv? M R)) 0]
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[(and (array-all-and (inline-array-map number-rational? M))
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(array-all-and (inline-array-map number-rational? R)))
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(define num (norm (matrix- M R)))
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(define den (norm R))
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(cond [(and (zero? num) (zero? den)) 0]
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[(zero? den) +inf.0]
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[else (assert (/ num den) nonnegative?)])]
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[else +inf.0]))
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;; ===================================================================================================
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;; Approximate predicates
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(: matrix-zero? (case-> ((Matrix Number) -> Boolean)
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((Matrix Number) Real -> Boolean)))
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(define (matrix-zero? M [eps (* 10 epsilon.0)])
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(cond [(eps . < . 0) (raise-argument-error 'matrix-identity? "Nonnegative-Real" 1 M eps)]
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[else
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(define-values (m n) (matrix-shape M))
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(<= (matrix-absolute-error M (make-matrix m n 0)) eps)]))
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(: matrix-identity? (case-> ((Matrix Number) -> Boolean)
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((Matrix Number) Real -> Boolean)))
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(define (matrix-identity? M [eps (* 10 epsilon.0)])
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(cond [(eps . < . 0) (raise-argument-error 'matrix-identity? "Nonnegative-Real" 1 M eps)]
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[else (and (square-matrix? M)
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(<= (matrix-relative-error M (identity-matrix (square-matrix-size M))) eps))]))
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(: matrix-orthonormal? (case-> ((Matrix Number) -> Boolean)
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((Matrix Number) Real -> Boolean)))
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(define (matrix-orthonormal? M [eps (* 10 epsilon.0)])
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(cond [(eps . < . 0) (raise-argument-error 'matrix-orthonormal? "Nonnegative-Real" 1 M eps)]
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[else (and (square-matrix? M)
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(matrix-identity? (matrix* M (matrix-hermitian M)) eps))]))
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