math.matrices: fix/rename mnorm, update all norms
closes #2244 - `mnorm` has been renamed to `normalize-matrix` to reflect what it actually does, which is normalize a matrix, not find a norm of a matrix. - `mnorm` is no longer a word defined here. - bugfix: previously, `normalize-matrix` found the supremum of a matrix (`mmax`), before taking the supremum's absolute value (`abs`) and dividing the matrix by it (`m/n`). for matrices containing only negative values and 0, the supremum is 0, and a `div-by-zero` error was thrown. `normalize-matrix` has been fixed to first `abs` all the matrix elements, and then find the supremum and divide, it also receieved a zero-matrix? guard for optimization and preventing `div-by-zero`. - new alias: `hilbert-schmidt-norm` for `frobenius-norm`, to go along with `math.matrices.extras.<hilbert-matrix>` and improve searchability by physicists. - new word: `matrix-p-norm`, written as an analogue of `math.vectors.p-norm`. - new word: `matrix-p-q-norm`, which generalizes entrywise matrix norm over the L^p,q vector space. - new word: `matrix-p-norm-entrywise`: `matrix-p-norm`'s fallback for p =/= 1, 2, inf; analogue of `math.vectors.p-norm-default`. - all norm words have gotten new docs, `zero-matrix?` guards as an optimisation, and most have gotten new tests.master
Cat Stevens
John Benediktsson
parent
2511fa72de
commit
573e4ed198
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@ -1,4 +1,4 @@
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! Copyright (C) 2005, 2010, 2018 Slava Pestov, Joe Groff, and Cat Stevens.
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! Copyright (C) 2005, 2010, 2018, 2020 Slava Pestov, Joe Groff, and Cat Stevens.
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USING: accessors arrays assocs generic.single formatting locals help.markup help.markup.private help.syntax io
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kernel math math.functions math.order math.ratios math.vectors opengl.gl prettyprint
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sequences sequences.generalizations urls ;
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@ -136,7 +136,15 @@ $nl
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} { $subsections
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main-diagonal
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anti-diagonal
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}
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"The following matrix norms are provided in the 𝑙ₚ vector space; these words are equivalent to ∥･∥ₚ for " { $snippet "p = 1, 2, ∞, n" } ", respectively:"
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{ $subsections
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m-1norm
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m-infinity-norm
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frobenius-norm
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matrix-p-norm-entrywise
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matrix-p-norm
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} ;
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! PREDICATE CLASSES
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@ -524,6 +532,17 @@ HELP: cols
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}
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} ;
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HELP: >square-matrix
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{ $values { "m" matrix } { "subset" square-matrix } }
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{ $description "Find only the " { $link2 square-matrix "square" } " subset of the input matrix." }
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{ $examples
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{ $example
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"USING: math.matrices prettyprint ;"
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"{ { 0 2 4 6 } { 1 3 5 7 } } >square-matrix ."
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"{ { 0 2 } { 1 3 } }"
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}
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} ;
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HELP: matrix-map
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{ $values { "matrix" matrix } { "quot" { $quotation ( ... elt -- ... elt' ) } } { "matrix'" matrix } }
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{ $description "Apply the quotation to every element of the matrix." }
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@ -938,7 +957,97 @@ HELP: mmax
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}
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} ;
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HELP: mnorm
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HELP: m-1norm
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{ $values { "m" matrix } { "n" number } }
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{ $description "Find the size of a matrix in 𝑙₁ (" { $snippet "L^₁" } ") vector space, usually written ∥･∥₁."
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$nl "This is the matrix norm when " { $snippet "p=1" } ", and is the overall maximum of the sums of the columns." }
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{ $notelist { $equiv-word-note "transpose" m-infinity-norm } }
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{ $examples
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{ $example
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"USING: math.matrices prettyprint ;"
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"{ { 2 -2 1 } { 1 3 -1 } { 2 -4 2 } } m-1norm ."
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"9"
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}
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} ;
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HELP: m-infinity-norm
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{ $values { "m" matrix } { "n" number } }
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{ $description "Find the size of a matrix, in 𝑙∞ (" { $snippet "L^∞" } ") vector space, usually written ∥･∥∞."
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$nl "This is the matrix norm when " { $snippet "p=∞" } ", and is the overall maximum of the sums of the rows." }
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{ $notelist { $equiv-word-note "transpose" m-1norm } }
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{ $examples
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{ $example
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"USING: math.matrices prettyprint ;"
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"{ { 2 -2 1 } { 1 3 -1 } { 2 -4 2 } } m-infinity-norm ."
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"8"
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}
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} ;
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HELP: frobenius-norm
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{ $values { "m" matrix } { "n" number } }
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{ $description "Find the size of a matrix in 𝑙₂ (" { $snippet "L^2" } ") vector space, usually written ∥･∥₂ₚ."
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$nl "This is the matrix norm when " { $snippet "p=2" } ", and is the square root of the sums of the squares of all the elements of the matrix." }
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{ $notelist
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{ "This norm is sometimes called the Hilbert-Schmidt norm." }
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{ "Because " $snippet { "p=2" } ", this word could be named " { $snippet "m-2norm" } "." }
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}
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{ $examples
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{ $example
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"USING: math.matrices prettyprint ;"
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"{ { 1 1 } { 1 1 } } frobenius-norm ."
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"2.0"
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}
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} ;
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{ m-1norm m-infinity-norm frobenius-norm } related-words
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HELP: matrix-p-q-norm
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{ $values { "m" matrix } { "p" "positive real number" } { "q" "positive real number" } { "n" "non-negative real number" } }
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{ $description "Find the size of a matrix in " { $snippet "L^p,q" } " vector space."
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$nl "This is the matrix norm for any " { $snippet "p, q ≥ 1" } ". It is still an entry-wise norm, like " { $link matrix-p-norm-entrywise } "." }
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{ $examples
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"Equivalent to " { $link frobenius-norm } " for " { $snippet "p = q = 2 " } ":"
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{ $example
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"USING: math.matrices prettyprint ;"
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"{ { 1 1 } { 1 1 } } 2 2 matrix-p-q-norm ."
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"2.0"
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}
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} ;
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HELP: matrix-p-norm-entrywise
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{ $values { "m" matrix } { "p" "positive real number" } { "n" "non-negative real number" } }
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{ $description "Find the entry-wise norm of a matrix, in 𝑙ₚ (" { $snippet "L^p" } ") vector space." }
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{ $notes "This word is distinct from a Schatten p-norm, as well as any of " { $links m-1norm frobenius-norm m-infinity-norm } "." }
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{ $examples
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{ $example
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"USING: math.matrices prettyprint ;"
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"4 4 1 <matrix> 2 matrix-p-norm-entrywise ."
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"4.0"
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}
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} ;
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HELP: matrix-p-norm
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{ $values { "m" matrix } { "p" "positive real number" } { "n" "non-negative real number" } }
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{ $description "Find the size of a matrix in 𝑙ₚ (" { $snippet "L^p" } ") vector space, usually written ∥･∥ₚ. For " { $snippet "p ≠ 1, 2, ∞" } ", this is an \"entry-wise\" norm." }
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{ $examples
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"Calls " { $link m-1norm } ":"
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{ $example
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"USING: math.matrices prettyprint ;"
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"4 4 1 <matrix> 1 matrix-p-norm ."
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"4"
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}
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"Falls back to " { $link matrix-p-norm-entrywise } ":"
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{ $example
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"USING: math.functions math.matrices prettyprint ;"
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"2 2 3 <matrix> 1.5 matrix-p-norm 7.559 10e-4 ~ ."
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"t"
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}
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} ;
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{ matrix-p-norm matrix-p-norm-entrywise } related-words
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{ matrix-p-norm matrix-p-q-norm } related-words
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HELP: normalize-matrix
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{ $values { "m" matrix } { "m'" matrix } }
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{ $description "Normalize a matrix. Each element from the input matrix is computed as a fraction of the maximum element. The maximum element becomes " { $snippet "1/1" } "." }
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{ $notelist
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@ -948,32 +1057,11 @@ HELP: mnorm
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{ $examples
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{ $example
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"USING: math.matrices prettyprint ;"
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"{ { 5 9 } { 15 17 } } mnorm ."
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"{ { 5 9 } { 15 17 } } normalize-matrix ."
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"{ { 5/17 9/17 } { 15/17 1 } }"
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}
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} ;
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HELP: m-infinity-norm
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{ $values { "m" matrix } { "n" number } } ;
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HELP: m-1norm
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{ $values { "m" matrix } { "n" number } } ;
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HELP: frobenius-norm
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{ $values { "m" matrix } { "n" number } }
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{ $notes "Also known as the Hilbert-Schmidt norm." } ;
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HELP: >square-matrix
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{ $values { "m" matrix } { "subset" square-matrix } }
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{ $description "Find only the " { $link2 square-matrix "square" } " subset of the input matrix." }
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{ $examples
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{ $example
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"USING: math.matrices prettyprint ;"
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"{ { 0 2 4 6 } { 1 3 5 7 } } >square-matrix ."
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"{ { 0 2 } { 1 3 } }"
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}
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} ;
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HELP: main-diagonal
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{ $values { "matrix" matrix } { "seq" sequence } }
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{ $description "Find the main diagonal of a matrix." $nl "This diagonal begins in the upper left of the matrix at index " { $snippet "{ 0 0 }" } ", continuing downward and rightward for all indices " { $snippet "{ n n }" } " in the " { $link square-matrix } " subset of the input (see " { $link <square-rows> } ")." }
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@ -1,4 +1,4 @@
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! Copyright (C) 2005, 2010, 2018 Slava Pestov, Joe Groff, and Cat Stevens.
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! Copyright (C) 2005, 2010, 2018, 2020 Slava Pestov, Joe Groff, and Cat Stevens.
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USING: arrays assocs combinators.short-circuit grouping kernel
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math math.statistics sequences sequences.deep tools.test ;
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IN: math.matrices
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@ -317,17 +317,6 @@ PRIVATE>
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mdot
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] unit-test
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! TODO: note: merge conflict from HEAD contained the following
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! ------------------------
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! predicates
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{ t } [ { } square-matrix? ] unit-test
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{ t } [ { { 1 } } square-matrix? ] unit-test
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{ t } [ { { 1 2 } { 3 4 } } square-matrix? ] unit-test
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{ f } [ { { 1 } { 2 3 } } square-matrix? ] unit-test
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{ f } [ { { 1 2 } } square-matrix? ] unit-test
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{ 9 }
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[ { { 2 -2 1 } { 1 3 -1 } { 2 -4 2 } } m-1norm ] unit-test
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@ -336,40 +325,37 @@ PRIVATE>
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{ 2.0 }
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[ { { 1 1 } { 1 1 } } frobenius-norm ] unit-test
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! from "intermediate commit"
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! any deep-empty matrix is null
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! it doesn't make any sense for { } to be null while { { } } to be considered nonnull
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{ t } [ {
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{ }
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{ { } }
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{ { { } } }
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{ { } { } { } }
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{ { { } } { { { } } } }
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} [ null-matrix? ] all?
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] unit-test
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{ f } [ {
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{ 1 2 }
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{ { 1 2 } }
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{ { 1 } { 2 } }
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{ { { 1 } } { 2 } { } }
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} [ null-matrix? ] any?
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] unit-test
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{ 10e-8 }
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[
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5.4772255
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{ { 1 2 } { 3 4 } } frobenius-norm
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] unit-test~
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{ t } [ 10 dup <zero-matrix> zero-matrix? ] unit-test
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{ t } [ 10 10 15 <simple-eye> zero-matrix? ] unit-test
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{ t } [ 0 dup <zero-matrix> null-matrix? ] unit-test
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{ f } [ 0 dup <zero-matrix> zero-matrix? ] unit-test
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{ f } [ 4 <identity-matrix> zero-matrix? ] unit-test
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{ 10e-6 }
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[
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36.94590
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{ { 1 2 } { 4 8 } { 16 32 } } frobenius-norm
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] unit-test~
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{ t } [ { } regular-matrix? ] unit-test
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{ t } [ { { } } regular-matrix? ] unit-test
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{ t } [ { { 1 2 } } regular-matrix? ] unit-test
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{ t } [ { { 1 2 } { 3 4 } } regular-matrix? ] unit-test
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{ t } [ { { 1 } { 3 } } regular-matrix? ] unit-test
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{ f } [ { { 1 2 } { 3 } } regular-matrix? ] unit-test
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{ f } [ { { 1 } { 3 2 } } regular-matrix? ] unit-test
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! TODO: note: lines since last HEAD comment were deleted in "fix more code and add more rigorous tests"
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! equivalent to frobenius for p = q = 2
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{ 2.0 }
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[ { { 1 1 } { 1 1 } } 2 2 matrix-p-q-norm ] unit-test
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{ 10e-7 }
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[
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33.456466
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{ { 1 2 } { 4 8 } { 16 32 } } 3 matrix-p-norm-entrywise
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] unit-test~
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{ { { -1 0 } { 0 0 } } }
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[ { { -2 0 } { 0 0 } } normalize-matrix ] unit-test
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{ { { -1 0 } { 0 1/2 } } }
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[ { { -2 0 } { 0 1 } } normalize-matrix ] unit-test
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{ t }
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[ 3 3 <zero-matrix> dup normalize-matrix = ] unit-test
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! diagonals
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@ -1,11 +1,11 @@
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! Copyright (C) 2005, 2010, 2018 Slava Pestov, Joe Groff, and Cat Stevens.
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! Copyright (C) 2005, 2010, 2018, 2020 Slava Pestov, Joe Groff, and Cat Stevens.
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! See http://factorcode.org/license.txt for BSD license.
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USING: accessors arrays classes.singleton columns combinators
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combinators.short-circuit combinators.smart formatting fry
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grouping kernel locals math math.bits math.functions math.order
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math.private math.ranges math.statistics math.vectors
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math.vectors.private sequences sequences.deep sequences.private
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slots.private summary ;
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math.vectors.private sequences sequences.deep sequences.extras
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sequences.private slots.private summary ;
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IN: math.matrices
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! defined here because of issue #1943
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@ -256,10 +256,48 @@ DEFER: matrix-set-nths
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: mmin ( m -- n ) [ 1/0. ] dip [ [ min ] each ] each ;
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: mmax ( m -- n ) [ -1/0. ] dip [ [ max ] each ] each ;
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: mnorm ( m -- m' ) dup mmax abs m/n ;
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: m-infinity-norm ( m -- n ) [ [ abs ] map-sum ] map supremum ;
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: m-1norm ( m -- n ) flip m-infinity-norm ;
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: frobenius-norm ( m -- n ) [ [ sq ] map-sum ] map-sum sqrt ;
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: m-infinity-norm ( m -- n )
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dup zero-matrix? [ drop 0. ] [
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[ [ abs ] map-sum ] map supremum
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] if ;
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: m-1norm ( m -- n )
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dup zero-matrix? [ drop 0. ] [
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flip m-infinity-norm
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] if ;
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: frobenius-norm ( m -- n )
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dup zero-matrix? [ drop 0. ] [
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[ [ sq ] map-sum ] map-sum sqrt
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] if ;
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ALIAS: hilbert-schmidt-norm frobenius-norm
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:: matrix-p-q-norm ( m p q -- n )
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m dup zero-matrix? [ drop 0. ] [
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[ [ sq ] map-sum q p / ^ ] map-sum q recip ^
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] if ;
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: matrix-p-norm-entrywise ( m p -- n )
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dup zero-matrix? [ 2drop 0. ] [
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[ flatten1 V{ } like ] dip p-norm-default
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] if ;
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: matrix-p-norm ( m p -- n )
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dup zero-matrix? [ 2drop 0. ] [
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{
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{ [ dup 1 = ] [ drop m-1norm ] }
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{ [ dup 2 = ] [ drop frobenius-norm ] }
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{ [ dup fp-infinity? ] [ drop m-infinity-norm ] }
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[ matrix-p-norm-entrywise ]
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} cond
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] if ;
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: normalize-matrix ( m -- m' )
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dup zero-matrix? [ ] [
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dup mabs mmax m/n
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] if ;
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! well-defined for square matrices; but works on nonsquare too
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: main-diagonal ( matrix -- seq )
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@ -49,9 +49,6 @@ ARTICLE: "math.matrices.extras" "Extra matrix operations"
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"Matrix algebra:"
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{ $subsections
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mmin
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mmax
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mnorm
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rank
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nullity
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