642 lines
21 KiB
Factor
642 lines
21 KiB
Factor
USING: arrays generic.single help.markup help.syntax kernel math
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math.matrices math.matrices.private math.matrices.extras
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math.order math.ratios math.vectors opengl.gl random sequences
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urls ;
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IN: math.matrices.extras
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ABOUT: "math.matrices.extras"
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ARTICLE: "math.matrices.extras" "Extra matrix operations"
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"These constructions have special mathematical properties:"
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{ $subsections
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<box-matrix>
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<hankel-matrix>
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<hilbert-matrix>
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<toeplitz-matrix>
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<vandermonde-matrix>
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}
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"Common transformation matrices:"
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{ $subsections
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<frustum-matrix4>
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<ortho-matrix4>
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<rotation-matrix3>
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<rotation-matrix4>
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<scale-matrix3>
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<scale-matrix4>
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<skew-matrix4>
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<translation-matrix4>
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<random-integer-matrix>
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<random-unit-matrix>
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}
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{ $subsections
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invertible-matrix?
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linearly-independent-matrix?
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}
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"Common algorithms on matrices:"
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{ $subsections
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gram-schmidt
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gram-schmidt-normalize
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kronecker-product
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outer-product
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}
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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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} { $subsections
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determinant 1/det m*1/det
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>minors >cofactors
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multiplicative-inverse
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}
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"Covariance in matrices:"
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{ $subsections
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covariance-matrix
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covariance-matrix-ddof
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sample-covariance-matrix
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}
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"Errors thrown by this vocabulary:"
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{ $subsections negative-power-matrix non-square-determinant undefined-inverse } ;
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HELP: invertible-matrix?
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{ $values { "matrix" matrix } { "?" boolean } }
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{ $description "Tests whether the input matrix has a " { $link multiplicative-inverse } ". In order for a matrix to be invertible, it must be a " { $link square-matrix } ", " { $emphasis "or" } ", if it is non-square, it must not be of " { $link +deficient-rank+ } "." }
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{ $examples { $example "USING: math.matrices.extras prettyprint ;" "" } } ;
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HELP: linearly-independent-matrix?
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{ $values { "matrix" matrix } { "?" boolean } }
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{ $description "Tests whether the input matrix is linearly independent." }
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{ $examples { $example "USING: math.matrices.extras prettyprint ;" "" } } ;
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! SINGLETON RANK TYPES
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HELP: rank-kind
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{ $class-description "The class of matrix rank quantifiers." } ;
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HELP: +full-rank+
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{ $class-description "A " { $link rank-kind } " describing a matrix of full rank." } ;
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HELP: +half-rank+
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{ $class-description "A " { $link rank-kind } " describing a matrix of half rank." } ;
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HELP: +zero-rank+
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{ $class-description "A " { $link rank-kind } " describing a matrix of zero rank." } ;
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HELP: +deficient-rank+
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{ $class-description "A " { $link rank-kind } " describing a matrix of deficient rank." } ;
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HELP: +uncalculated-rank+
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{ $class-description "A " { $link rank-kind } " describing a matrix whose rank is not (yet) known." } ;
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! ERRORS
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HELP: negative-power-matrix
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{ $values { "m" matrix } { "n" integer } }
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{ $description "Throws a " { $link negative-power-matrix } " error." }
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{ $error-description "Given the semantics of " { $link m^n } ", negative exponents are not within the domain of the power matrix function." } ;
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HELP: non-square-determinant
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{ $values { "m" integer } { "n" integer } }
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{ $description "Throws a " { $link non-square-determinant } " error." }
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{ $error-description { $link determinant } " was used with a non-square matrix whose dimensions are " { $snippet "m x n" } ". It is not generally possible to find the determinant of a non-square matrix." } ;
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HELP: undefined-inverse
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{ $values { "m" integer } { "n" integer } { "r" rank-kind } }
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{ $description "Throws an " { $link undefined-inverse } " error." }
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{ $error-description { $link multiplicative-inverse } " was used with a non-square matrix of rank " { $snippet "rank" } " whose dimensions are " { $snippet "m x n" } ". It is not generally possible to find the inverse of a " { $link +deficient-rank+ } " non-square " { $link matrix } "." } ;
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HELP: <random-integer-matrix>
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{ $values { "m" integer } { "n" integer } { "max" integer } { "matrix" matrix } }
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{ $description "Creates a " { $snippet "m x n" } " " { $link matrix } " full of random, possibly signed " { $link integer } "s whose absolute values are less than or equal to " { $snippet "max" } ", as given by " { $link random-integers } "." }
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{ $notelist
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{ "The signedness of the numbers in the resulting matrix will be randomized. Use " { $link mabs } " with this word to generate a matrix of random positive integers." }
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{ $equiv-word-note "integral" <random-unit-matrix> }
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}
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{ $errors { $link no-method } " if " { $snippet "max"} " is not an " { $link integer } "." }
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{ $examples
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{ $unchecked-example
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"USING: math.matrices.extras prettyprint ;"
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"2 4 15 <random-integer-matrix> ."
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"{ { -9 -9 1 3 } { -14 -8 14 10 } }"
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}
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} ;
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HELP: <random-unit-matrix>
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{ $values { "m" integer } { "n" integer } { "max" number } { "matrix" matrix } }
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{ $description "Creates a " { $snippet "m x n" } " " { $link matrix } " full of random, possibly signed " { $link float } "s as a fraction of " { $snippet "max" } "." }
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{ $notelist
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{ "The signedness of the numbers in the resulting matrix will be randomized. Use " { $link mabs } " with this word to generate a matrix of random positive numbers." }
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{ $equiv-word-note "real" <random-integer-matrix> }
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{ "This word is implemented by generating sub-integral floats through " { $link random-units } " and multiplying by random integers less than or equal to " { $snippet "max" } "." }
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}
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{ $examples
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{ $unchecked-example
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"USING: math.matrices.extras prettyprint ;"
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"4 2 15 <random-unit-matrix> ."
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"{
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{ -3.713295909201797 3.815787135075961 }
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{ -2.460506890603817 1.535222788710546 }
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{ 3.692213981267878 -1.462963244399762 }
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{ 13.8967592095433 -6.688509969360172 }
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}"
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}
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} ;
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HELP: <hankel-matrix>
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{ $values { "n" integer } { "matrix" matrix } }
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{ $description
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"A Hankel matrix is a symmetric, " { $link square-matrix } " in which each ascending skew-diagonal from left to right is constant. See " { $url URL" https://en.wikipedia.org/wiki/Hankel_matrix" "hankel matrix" } "."
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$nl
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"The following is true of any Hankel matrix" { $snippet "A" } ": " { $snippet "A[i][j] = A[j][i] = a[i+j-2]" } "."
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$nl
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"The " { $link <toeplitz-matrix> } " is an upside-down Hankel matrix."
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$nl
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"The " { $link <hilbert-matrix> } " is a special case of the Hankel matrix."
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}
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{ $examples
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{ $example
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"USING: math.matrices.extras prettyprint ;"
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"4 <hankel-matrix> ."
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"{ { 1 2 3 4 } { 2 3 4 0 } { 3 4 0 0 } { 4 0 0 0 } }"
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}
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} ;
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HELP: <hilbert-matrix>
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{ $values { "m" integer } { "n" integer } { "matrix" matrix } }
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{ $description
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"A Hilbert matrix is a " { $link square-matrix } " " { $snippet "A" } " in which entries are the unit fractions "
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{ $snippet "A[i][j] = 1/(i+j-1)" }
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". See " { $url URL" https://en.wikipedia.org/wiki/Hilbert_matrix" "hilbert matrix" } "."
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$nl
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"A Hilbert matrix is a special case of the " { $link <hankel-matrix> } "."
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}
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{ $examples
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{ $example
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"USING: math.matrices.extras prettyprint ;"
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"1 2 <hilbert-matrix> ."
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"{ { 1 1/2 } }"
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}
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{ $example
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"USING: math.matrices.extras prettyprint ;"
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"3 6 <hilbert-matrix> ."
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"{
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{ 1 1/2 1/3 1/4 1/5 1/6 }
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{ 1/2 1/3 1/4 1/5 1/6 1/7 }
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{ 1/3 1/4 1/5 1/6 1/7 1/8 }
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}"
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}
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} ;
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HELP: <toeplitz-matrix>
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{ $values { "n" integer } { "matrix" matrix } }
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{ $description "A Toeplitz matrix is an upside-down " { $link <hankel-matrix> } ". Unlike the Hankel matrix, a Toeplitz matrix can be non-square. See " { $url URL" https://en.wikipedia.org/wiki/Hankel_matrix" "hankel matrix" } "."
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}
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{ $examples
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{ $example
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"USING: math.matrices.extras prettyprint ;"
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"4 <toeplitz-matrix> ."
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"{ { 1 2 3 4 } { 2 1 2 3 } { 3 2 1 2 } { 4 3 2 1 } }"
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}
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} ;
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HELP: <box-matrix>
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{ $values { "r" integer } { "matrix" matrix } }
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{ $description "Create a box matrix (a " { $link square-matrix } ") with the dimensions of " { $snippet "r x r" } ", filled with ones. The number of elements in the output scales linearly (" { $snippet "(r*2)+1" } ") with " { $snippet "r" } "." }
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{ $examples
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{ $example
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"USING: math.matrices.extras prettyprint ;"
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"2 <box-matrix> ."
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"{
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{ 1 1 1 1 1 }
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{ 1 1 1 1 1 }
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{ 1 1 1 1 1 }
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{ 1 1 1 1 1 }
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{ 1 1 1 1 1 }
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}"
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}
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{ $example
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"USING: math.matrices.extras prettyprint ;"
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"3 <box-matrix> ."
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"{
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{ 1 1 1 1 1 1 1 }
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{ 1 1 1 1 1 1 1 }
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{ 1 1 1 1 1 1 1 }
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{ 1 1 1 1 1 1 1 }
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{ 1 1 1 1 1 1 1 }
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{ 1 1 1 1 1 1 1 }
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{ 1 1 1 1 1 1 1 }
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}"
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}
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} ;
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HELP: <scale-matrix3>
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{ $values { "factors" sequence } { "matrix" matrix } }
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{ $description "Make a " { $snippet "3 x 3" } " scaling matrix, used to scale an object in 3 dimensions. See " { $url URL" https://en.wikipedia.org/wiki/Scaling_(geometry)#Matrix_representation" "scaling matrix on Wikipedia" } "." }
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{ $notelist
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{ $finite-input-note "three" "factors" }
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{ $equiv-word-note "3-matrix" <scale-matrix4> }
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}
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{ $examples
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{ $example
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"USING: math.matrices.extras prettyprint ;"
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"{ 22 33 -44 } <scale-matrix4> ."
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"{
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{ 22 0.0 0.0 0.0 }
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{ 0.0 33 0.0 0.0 }
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{ 0.0 0.0 -44 0.0 }
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{ 0.0 0.0 0.0 1.0 }
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}"
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}
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} ;
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HELP: <scale-matrix4>
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{ $values { "factors" sequence } { "matrix" matrix } }
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{ $description "Make a " { $snippet "4 x 4" } " scaling matrix, used to scale an object in 3 or more dimensions. See " { $url URL" https://en.wikipedia.org/wiki/Scaling_(geometry)#Matrix_representation" "scaling matrix on Wikipedia" } "." }
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{ $notelist
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{ $finite-input-note "three" "factors" }
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{ $equiv-word-note "4-matrix" <scale-matrix3> }
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}
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{ $examples
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{ $example
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"USING: math.matrices.extras prettyprint ;"
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"{ 22 33 -44 } <scale-matrix4> ."
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"{
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{ 22 0.0 0.0 0.0 }
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{ 0.0 33 0.0 0.0 }
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{ 0.0 0.0 -44 0.0 }
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{ 0.0 0.0 0.0 1.0 }
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}"
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}
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} ;
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HELP: <ortho-matrix4>
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{ $values { "factors" sequence } { "matrix" matrix } }
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{ $description "Create a " { $link <scale-matrix4> } ", with the scale factors inverted." }
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{ $notelist
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{ $finite-input-note "three" "factors" }
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{ $equiv-word-note "inverse" <scale-matrix4> }
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}
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{ $examples
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{ $example
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"USING: math.matrices.extras prettyprint ;"
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"{ -9.3 100 1/2 } <ortho-matrix4> ."
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"{
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{ -0.1075268817204301 0.0 0.0 0.0 }
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{ 0.0 1/100 0.0 0.0 }
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{ 0.0 0.0 2 0.0 }
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{ 0.0 0.0 0.0 1.0 }
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}"
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}
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} ;
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HELP: <frustum-matrix4>
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{ $values { "xy-dim" pair } { "near" number } { "far" number } { "matrix" matrix } }
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{ $description "Make a " { $snippet "4 x 4" } " matrix suitable for representing an occlusion frustum. A viewing or occlusion frustum is the three-dimensional region of a three-dimensional object which is visible on the screen. See " { $url URL" https://en.wikipedia.org/wiki/Frustum" "frustum on Wikipedia" } "." }
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{ $notes { $finite-input-note "two" "xy-dim" } }
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{ $examples
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{ $example
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"USING: math.matrices.extras prettyprint ;"
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"{ 5 4 } 5 6 <frustum-matrix4> ."
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"{
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{ 1.0 0.0 0.0 0.0 }
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{ 0.0 1.25 0.0 0.0 }
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{ 0.0 0.0 -11.0 -60.0 }
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{ 0.0 0.0 -1.0 0.0 }
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}"
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}
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} ;
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{ <frustum-matrix4> glFrustum } related-words
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HELP: cartesian-matrix-map
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{ $values { "matrix" matrix } { "quot" { $quotation ( ... pair matrix -- ... matrix' ) } } { "matrix-seq" { $sequence matrix } } }
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{ $description "Calls the quotation with the matrix and the coordinate pair of the current element on the stack, with the matrix on the top of the stack." }
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{ $examples
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{ $example
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"USING: arrays math.matrices.extras prettyprint ;"
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"{ { 21 22 } { 23 24 } } [ 2array ] cartesian-matrix-map ."
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"{
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{
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{ { 0 0 } { { 21 22 } { 23 24 } } }
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{ { 0 1 } { { 21 22 } { 23 24 } } }
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}
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{
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{ { 1 0 } { { 21 22 } { 23 24 } } }
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{ { 1 1 } { { 21 22 } { 23 24 } } }
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}
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}"
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}
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}
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{ $notelist
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{ $equiv-word-note "orthogonal" cartesian-column-map }
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{ $equiv-word-note "two-dimensional" map-index }
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$2d-only-note
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} ;
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HELP: cartesian-column-map
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{ $values { "matrix" matrix } { "quot" { $quotation ( ... pair matrix -- ... matrix' ) } } { "matrix-seq" { $sequence matrix } } }
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{ $notelist
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{ $equiv-word-note "orthogonal" cartesian-matrix-map }
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$2d-only-note
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} ;
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HELP: gram-schmidt
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{ $values { "matrix" matrix } { "orthogonal" matrix } }
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{ $description "Apply a Gram-Schmidt transform on the matrix." }
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{ $examples
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{ $example
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"USING: math.matrices.extras prettyprint ;"
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"{ { 1 2 } { 3 4 } { 5 6 } } gram-schmidt ."
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"{ { 1 2 } { 4/5 -2/5 } { 0 0 } }"
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}
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} ;
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HELP: gram-schmidt-normalize
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{ $values { "matrix" matrix } { "orthonormal" matrix } }
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{ $description "Apply a Gram-Schmidt transform on the matrix, and " { $link normalize } " each row of the result, resulting in an orthogonal and normalized matrix (orthonormal)." }
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{ $examples
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{ $example
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"USING: math.matrices.extras prettyprint ;"
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"{ { 1 2 } { 3 4 } { 5 6 } } gram-schmidt-normalize ."
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"{
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{ 0.4472135954999579 0.8944271909999159 }
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{ 0.894427190999916 -0.447213595499958 }
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{ NAN: 8000000000000 NAN: 8000000000000 }
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}"
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}
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} ;
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HELP: m^n
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{ $values { "m" matrix } { "n" object } }
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{ $description "Compute the " { $snippet "nth" } " power of the input matrix. If " { $snippet "n" } " is " { $snippet "-1" } ", the inverse of the matrix is calculated (but see " { $link multiplicative-inverse } " for pitfalls)." }
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{ $errors
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{ $link negative-power-matrix } " if " { $snippet "n" } " is a negative number other than " { $snippet "-1" } "."
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$nl
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{ $link undefined-inverse } " if " { $snippet "n" } " is " { $snippet "-1" } " and the " { $link multiplicative-inverse } " of " { $snippet "m" } " is undefined."
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}
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{ $notelist
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{ $equiv-word-note "swapped" n^m }
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$2d-only-note
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{ $matrix-scalar-note max abs / }
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}
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{ $examples
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{ $example
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"USING: math.matrices.extras prettyprint ;"
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"{ { 1 2 } { 3 4 } } 2 m^n ."
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"{ { 7 10 } { 15 22 } }"
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}
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} ;
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HELP: n^m
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{ $values { "n" object } { "m" matrix } }
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{ $description "Because it is nonsensical to raise a number to the power of a matrix, this word exists to save typing " { $snippet "swap m^n" } ". See " { $link m^n } " for more information." }
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{ $errors
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{ $link negative-power-matrix } " if " { $snippet "n" } " is a negative number other than " { $snippet "-1" } "."
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$nl
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{ $link undefined-inverse } " if " { $snippet "n" } " is " { $snippet "-1" } " and the " { $link multiplicative-inverse } " of " { $snippet "m" } " is undefined."
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}
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{ $notelist
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{ $equiv-word-note "swapped" m^n }
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$2d-only-note
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{ $matrix-scalar-note max abs / }
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}
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{ $examples
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{ $example
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"USING: math.matrices.extras prettyprint ;"
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"2 { { 1 2 } { 3 4 } } n^m ."
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"{ { 7 10 } { 15 22 } }"
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}
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} ;
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HELP: kronecker-product
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{ $values { "m1" matrix } { "m2" matrix } { "m" matrix } }
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{ $description "Calculates the " { $url URL" http://enwp.org/Kronecker_product" "Kronecker product" } " of two matrices. This product can be described as a generalization of the vector-based " { $link outer-product } " to matrices. The Kronecker product gives the matrix of the tensor product with respect to a standard choice of basis." }
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{ $notelist
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{ $equiv-word-note "matrix" outer-product }
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$2d-only-note
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{ $matrix-scalar-note * }
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}
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{ $examples
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{ $unchecked-example
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"USING: math.matrices.extras prettyprint ;"
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"{
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{ 1 2 }
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{ 3 4 }
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} {
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{ 0 5 }
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{ 6 7 }
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} kronecker-product ."
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"{
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{ 0 5 0 10 }
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{ 6 7 12 14 }
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{ 0 15 0 20 }
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{ 18 21 24 28 }
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}" }
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} ;
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HELP: outer-product
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{ $values { "u" sequence } { "v" sequence } { "matrix" matrix } }
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{ $description "Computes the " { $url URL" http:// enwp.org/Outer_product" "outer-product product" } " of " { $snippet "u" } " and " { $snippet "v" } "." }
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{ $examples
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{ $example
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"USING: math.matrices.extras prettyprint ;"
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"{ 5 6 7 } { 1 2 3 } outer-product ."
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"{ { 5 10 15 } { 6 12 18 } { 7 14 21 } }" }
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} ;
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HELP: rank
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{ $values { "matrix" matrix } { "rank" rank-kind } }
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{ $contract "The " { $emphasis "rank" } " of a " { $link matrix } " is how its number of linearly independent columns compare to the maximal number of linearly independent columns for a matrix with the same dimension." }
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{ $notes "See " { $url "https://en.wikipedia.org/wiki/Rank_(linear_algebra)" } " for more information." } ;
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HELP: nullity
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{ $values { "matrix" matrix } { "nullity" rank-kind } }
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;
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HELP: determinant
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{ $values { "matrix" square-matrix } { "determinant" number } }
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{ $contract "Compute the determinant of the input matrix. Generally, the determinant of a matrix is a scaling factor of the transformation described by the matrix." }
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{ $notelist
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$2d-only-note
|
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{ $matrix-scalar-note max - * }
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}
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{ $errors { $link non-square-determinant } " if the input matrix is not a " { $link square-matrix } "." }
|
|
{ $examples
|
|
{ $example
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|
"USING: math.matrices.extras prettyprint ;"
|
|
"{
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|
{ 3 0 -1 }
|
|
{ -3 1 3 }
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{ 2 -5 4 }
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|
} determinant ."
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|
"44"
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|
}
|
|
{ $example
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|
"USING: math.matrices.extras prettyprint ;"
|
|
"{
|
|
{ -8 -8 13 11 10 -5 -14 }
|
|
{ 3 -11 -8 3 -7 -3 4 }
|
|
{ 10 4 -5 3 0 -6 -12 }
|
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{ -14 0 -3 -8 10 0 10 }
|
|
{ 3 -6 1 -10 -9 10 0 }
|
|
{ 5 -12 -14 6 5 -1 -7 }
|
|
{ -9 -14 -8 5 2 2 -2 }
|
|
} determinant ."
|
|
"-103488155"
|
|
}
|
|
} ;
|
|
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HELP: 1/det
|
|
{ $values { "matrix" square-matrix } { "1/det" number } }
|
|
{ $description "Find the inverse (" { $link recip } ") of the " { $link determinant } " of the input matrix." }
|
|
{ $notelist
|
|
$2d-only-note
|
|
{ $matrix-scalar-note determinant recip }
|
|
}
|
|
{ $errors
|
|
{ $link non-square-determinant } " if the input matrix is not a " { $link square-matrix } "."
|
|
$nl
|
|
{ $link division-by-zero } " if the " { $link determinant } " of the input matrix is " { $snippet "0" } "."
|
|
}
|
|
{ $examples
|
|
{ $example
|
|
"USING: math.matrices.extras prettyprint ;"
|
|
"{
|
|
{ 0 10 -12 4 }
|
|
{ -9 6 -11 9 }
|
|
{ -5 -10 0 2 }
|
|
{ -7 -11 10 11 }
|
|
} 1/det ."
|
|
"-1/9086"
|
|
}
|
|
} ;
|
|
|
|
HELP: m*1/det
|
|
{ $values { "matrix" square-matrix } { "matrix'" square-matrix } }
|
|
{ $description "Multiply the input matrix by the inverse (" { $link recip } ") of its " { $link determinant } "." }
|
|
{ $notelist
|
|
{ "This word is used to implement " { $link recip } " for " { $link square-matrix } "." }
|
|
$2d-only-note
|
|
{ $matrix-scalar-note determinant recip }
|
|
}
|
|
{ $examples
|
|
{ $example
|
|
"USING: math.matrices.extras prettyprint ;"
|
|
"{
|
|
{ -14 0 -13 7 }
|
|
{ -4 11 7 -12 }
|
|
{ -3 2 9 -14 }
|
|
{ 3 -5 10 -2 }
|
|
} m*1/det ."
|
|
"{
|
|
{ 7/6855 0 13/13710 -7/13710 }
|
|
{ 2/6855 -11/13710 -7/13710 2/2285 }
|
|
{ 1/4570 -1/6855 -3/4570 7/6855 }
|
|
{ -1/4570 1/2742 -1/1371 1/6855 }
|
|
}"
|
|
}
|
|
}
|
|
;
|
|
|
|
HELP: >minors
|
|
{ $values { "matrix" square-matrix } { "matrix'" square-matrix } }
|
|
{ $description "Calculate the " { $emphasis "matrix of minors" } " of the input matrix. See " { $url URL" https://en.wikipedia.org/wiki/Minor_(linear_algebra)" "minor on Wikipedia" } "." }
|
|
{ $notelist
|
|
$keep-shape-note
|
|
$2d-only-note
|
|
{ $matrix-scalar-note determinant }
|
|
}
|
|
{ $errors { $link non-square-determinant } " if the input matrix is not a " { $link square-matrix } "." }
|
|
{ $examples
|
|
{ $example
|
|
"USING: math.matrices.extras prettyprint ;"
|
|
"{
|
|
{ -8 0 7 -11 }
|
|
{ 15 0 -3 -11 }
|
|
{ 1 -10 -4 6 }
|
|
{ 11 -15 3 -15 }
|
|
} >minors ."
|
|
"{
|
|
{ 1710 -130 2555 -1635 }
|
|
{ -690 -286 -2965 1385 }
|
|
{ 1650 -754 3795 -1215 }
|
|
{ 1100 416 2530 -810 }
|
|
}"
|
|
}
|
|
} ;
|
|
|
|
HELP: >cofactors
|
|
{ $values { "matrix" matrix } { "matrix'" matrix } }
|
|
{ $description "Calculate the " { $emphasis "matrix of cofactors" } " of the input matrix. See " { $url URL" https://en.wikipedia.org/wiki/Minor_(linear_algebra)#Inverse_of_a_matrix" "matrix of cofactors on Wikipedia" } ". Alternating elements of the input matrix have their signs inverted." $nl "On odd rows, the even elements have their signs inverted. On even rows, odd elements have their signs inverted." }
|
|
{ $notelist
|
|
$keep-shape-note
|
|
$2d-only-note
|
|
{ $matrix-scalar-note neg }
|
|
}
|
|
{ $examples
|
|
{ $example
|
|
"USING: math.matrices.extras prettyprint ;"
|
|
"{
|
|
{ 8 0 7 11 }
|
|
{ 15 0 3 11 }
|
|
{ 1 10 4 6 }
|
|
{ 11 15 3 15 }
|
|
} >cofactors ."
|
|
"{
|
|
{ 8 0 7 -11 }
|
|
{ -15 0 -3 11 }
|
|
{ 1 -10 4 -6 }
|
|
{ -11 15 -3 15 }
|
|
}"
|
|
}
|
|
{ $example
|
|
"USING: math.matrices.extras prettyprint ;"
|
|
"{
|
|
{ -8 0 7 -11 }
|
|
{ 15 0 -3 -11 }
|
|
{ 1 -10 -4 6 }
|
|
{ 11 -15 3 -15 }
|
|
} >cofactors ."
|
|
"{
|
|
{ -8 0 7 11 }
|
|
{ -15 0 3 -11 }
|
|
{ 1 10 -4 -6 }
|
|
{ -11 -15 -3 -15 }
|
|
}"
|
|
}
|
|
} ;
|
|
|
|
HELP: multiplicative-inverse
|
|
{ $values { "x" matrix } { "y" matrix } }
|
|
{ $description "Calculate the multiplicative inverse of the input." $nl "If the input is a " { $link square-matrix } ", this is done by multiplying the " { $link transpose } " of the " { $link2 >cofactors "cofactors" } " of the " { $link2 >minors "minors" } " of the input matrix by the " { $link2 1/det "inverse of the determinant" } " of the input matrix." }
|
|
{ $notelist
|
|
$keep-shape-note
|
|
$2d-only-note
|
|
{ $matrix-scalar-note determinant >cofactors 1/det }
|
|
}
|
|
{ $errors { $link non-square-determinant } " if the input matrix is not a " { $link square-matrix } "." } ;
|
|
|
|
|
|
HELP: covariance-matrix-ddof
|
|
{ $values { "matrix" matrix } { "ddof" object } { "cov" matrix } }
|
|
;
|
|
HELP: covariance-matrix
|
|
{ $values { "matrix" matrix } { "cov" matrix } }
|
|
;
|
|
HELP: sample-covariance-matrix
|
|
{ $values { "matrix" matrix } { "cov" matrix } }
|
|
;
|
|
HELP: population-covariance-matrix
|
|
{ $values { "matrix" matrix } { "cov" matrix } }
|
|
;
|