! Copyright (C) 2005, 2010, 2018, 2020 Slava Pestov, Joe Groff, and Cat Stevens.
-USING: accessors arrays assocs generic.single formatting locals help.markup help.markup.private help.syntax io
-kernel math math.functions math.order math.ratios math.vectors opengl.gl prettyprint
-sequences sequences.generalizations urls ;
+USING: arrays assocs help.markup help.markup.private help.syntax
+kernel math math.functions math.order math.vectors sequences
+sequences.generalizations ;
IN: math.matrices
<PRIVATE
$nl
"In this vocabulary's documentation, " { $snippet "m" } " and " { $snippet "matrix" } " are the conventional names used for a given matrix object. " { $snippet "m" } " may also refer to a number."
$nl
-"The " { $vocab-link "math.matrices.extras" } "vocabulary implements extensions to this one."
+"The " { $vocab-link "math.matrices.extras" } " vocabulary implements extensions to this one."
$nl
"Matrices are classified their mathematical properties, and by predicate words:"
$nl
zero-matrix?
zero-square-matrix?
null-matrix?
-
}
"There are many ways to create 2-dimensional matrices:"
}
"By-element mathematical operations on a matrix:"
-{ $subsections mneg m+n m-n m*n m/n n+m n-m n*m n/m }
+{ $subsections matrix-normalize mneg m+n m-n m*n m/n n+m n-m n*m n/m }
-"By-element mathematical operations of two matricess:"
+"By-element mathematical operations of two matrices:"
{ $subsections m+ m- m* m/ m~ }
"Dot product (multiplication) of vectors and matrices:"
anti-diagonal
}
-"The following matrix norms are provided in the ๐โ vector space; these words are equivalent to โฅ๏ฝฅโฅโ for " { $snippet "p = 1, 2, โ, n" } ", respectively:"
+"The following matrix norms are provided in the ๐โ and " { $snippet "L^p,q" } " vector spaces; these words are equivalent to โฅ๏ฝฅโฅโ and โฅ๏ฝฅโฅ^p,q for " { $snippet "p = 1, 2, โ, โ" } ", and " { $snippet "p, q โ โ" } ", respectively:"
{ $subsections
- m-1norm
- m-infinity-norm
- frobenius-norm
- matrix-p-norm-entrywise
+ matrix-l1-norm
+ matrix-l2-norm
+ matrix-l-infinity-norm
matrix-p-norm
+ matrix-p-q-norm
+}
+"For readability, user code should prefer the available generic versions of the above, from " { $vocab-link "math.vectors" } ", which are optimized the same:"
+{ $subsections
+ l1-norm l2-norm l-infinity-norm p-norm
} ;
! PREDICATE CLASSES
HELP: <diagonal-matrix>
{ $values { "diagonal-seq" sequence } { "matrix" matrix } }
-{ $description "Creates a matrix with the specified main diagonal. This word has the opposite effect of " { $link anti-diagonal } "." }
+{ $description "Creates a matrix with the specified main diagonal. This word has the opposite effect of " { $link main-diagonal } "." }
{ $notes "To use a diagonal starting in the lower right, reverse the input sequence before calling this word." }
{ $examples
{ $example
HELP: <anti-diagonal-matrix>
{ $values { "diagonal-seq" sequence } { "matrix" matrix } }
-{ $description "Creates a matrix with the specified anti-diagonal. This word has the opposite effect of " { $link main-diagonal } "." }
+{ $description "Creates a matrix with the specified anti-diagonal. This word has the opposite effect of " { $link anti-diagonal } "." }
{ $notes "To use a diagonal starting in the lower left, reverse the input sequence before calling this word." }
{ $examples
{ $example
}
} ;
-HELP: m-1norm
+{ l2-norm frobenius-norm hilbert-schmidt-norm } related-words
+
+HELP: matrix-l1-norm
{ $values { "m" matrix } { "n" number } }
-{ $description "Find the size of a matrix in ๐โ (" { $snippet "L^โ" } ") vector space, usually written โฅ๏ฝฅโฅโ."
+{ $description "Find the norm (size) of a matrix in ๐โ (" { $snippet "L^โ" } ") vector space, usually written โฅ๏ฝฅโฅโ."
$nl "This is the matrix norm when " { $snippet "p=1" } ", and is the overall maximum of the sums of the columns." }
-{ $notelist { $equiv-word-note "transpose" m-infinity-norm } }
+{ $notelist
+ { "User code should call the generic " { $link l1-norm } " instead." }
+ { $equiv-word-note "matrix-specific" l1-norm }
+ { $equiv-word-note { $snippet "p = 1" } matrix-p-norm }
+ { $equiv-word-note "transpose" matrix-l-infinity-norm }
+ $2d-only-note
+}
{ $examples
{ $example
"USING: math.matrices prettyprint ;"
- "{ { 2 -2 1 } { 1 3 -1 } { 2 -4 2 } } m-1norm ."
+ "{ { 2 -2 1 } { 1 3 -1 } { 2 -4 2 } } matrix-l1-norm ."
"9"
}
} ;
-HELP: m-infinity-norm
+HELP: matrix-l2-norm
{ $values { "m" matrix } { "n" number } }
-{ $description "Find the size of a matrix, in ๐โ (" { $snippet "L^โ" } ") vector space, usually written โฅ๏ฝฅโฅโ."
-$nl "This is the matrix norm when " { $snippet "p=โ" } ", and is the overall maximum of the sums of the rows." }
-{ $notelist { $equiv-word-note "transpose" m-1norm } }
+{ $description "Find the norm (size) of a matrix in ๐โ (" { $snippet "L^2" } ") vector space, usually written โฅ๏ฝฅโฅโ."
+$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." }
+{ $notelist
+ { "This norm is sometimes called the Hilbert-Schmidt norm." }
+ { "User code should call the generic " { $link p-norm } " instead." }
+ { $equiv-word-note "matrix-specific" l2-norm }
+ { $equiv-word-note { $snippet "p = 2" } matrix-p-norm }
+ { $equiv-word-note "transpose" l1-norm }
+ $2d-only-note
+}
{ $examples
{ $example
"USING: math.matrices prettyprint ;"
- "{ { 2 -2 1 } { 1 3 -1 } { 2 -4 2 } } m-infinity-norm ."
- "8"
+ "{ { 1 1 } { 1 1 } } matrix-l2-norm ."
+ "2.0"
}
} ;
-HELP: frobenius-norm
+HELP: matrix-l-infinity-norm
{ $values { "m" matrix } { "n" number } }
-{ $description "Find the size of a matrix in ๐โ (" { $snippet "L^2" } ") vector space, usually written โฅ๏ฝฅโฅโโ."
-$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." }
+{ $description "Find the norm (size) of a matrix, in ๐โ (" { $snippet "L^โ" } ") vector space, usually written โฅ๏ฝฅโฅโ."
+$nl "This is the matrix norm when " { $snippet "p=โ" } ", and is the overall maximum of the sums of the rows." }
{ $notelist
- { "This norm is sometimes called the Hilbert-Schmidt norm." }
- { "Because " $snippet { "p=2" } ", this word could be named " { $snippet "m-2norm" } "." }
+ { "User code should call the generic " { $link l1-norm } " instead." }
+ { $equiv-word-note "matrix-specific" l-infinity-norm }
+ { $equiv-word-note { $snippet "p = โ" } matrix-p-norm }
+ { $equiv-word-note "transpose" matrix-l1-norm }
+ $2d-only-note
}
{ $examples
{ $example
"USING: math.matrices prettyprint ;"
- "{ { 1 1 } { 1 1 } } frobenius-norm ."
- "2.0"
+ "{ { 2 -2 1 } { 1 3 -1 } { 2 -4 2 } } matrix-l-infinity-norm ."
+ "8"
}
} ;
-{ m-1norm m-infinity-norm frobenius-norm } related-words
-
HELP: matrix-p-q-norm
-{ $values { "m" matrix } { "p" "positive real number" } { "q" "positive real number" } { "n" "non-negative real number" } }
-{ $description "Find the size of a matrix in " { $snippet "L^p,q" } " vector space."
-$nl "This is the matrix norm for any " { $snippet "p, q รข\89ยฅ 1" } ". It is still an entry-wise norm, like " { $link matrix-p-norm-entrywise } "." }
+{ $values { "m" matrix } { "p" "a positive real number" } { "q" "a positive real number" } { "n" "a non-negative real number" } }
+{ $description "Find the norm (size) of a matrix in " { $snippet "L^p,q" } " vector space."
+$nl "This is the matrix norm for any " { $snippet "p, q รข\88\88 รข\84\9d" } ". It is still an entry-wise norm, like " { $link matrix-p-norm-entrywise } ", and is not an induced or Schatten norm." }
{ $examples
- "Equivalent to " { $link frobenius-norm } " for " { $snippet "p = q = 2 " } ":"
+ "Equivalent to " { $link l2-norm } " for " { $snippet "p = q = 2 " } ":"
{ $example
"USING: math.matrices prettyprint ;"
"{ { 1 1 } { 1 1 } } 2 2 matrix-p-q-norm ."
} ;
HELP: matrix-p-norm-entrywise
-{ $values { "m" matrix } { "p" "positive real number" } { "n" "non-negative real number" } }
+{ $values { "m" matrix } { "p" "a positive real number" } { "n" "a non-negative real number" } }
{ $description "Find the entry-wise norm of a matrix, in ๐โ (" { $snippet "L^p" } ") vector space." }
-{ $notes "This word is distinct from a Schatten p-norm, as well as any of " { $links m-1norm frobenius-norm m-infinity-norm } "." }
+{ $notes "This word is not an induced or Schatten norm, and it is distinct from all of " { $links matrix-l1-norm matrix-l2-norm matrix-l-infinity-norm } "." }
{ $examples
- { $example
- "USING: math.matrices prettyprint ;"
- "4 4 1 <matrix> 2 matrix-p-norm-entrywise ."
- "4.0"
- }
+ { $example
+ "USING: math.matrices prettyprint ;"
+ "4 4 1 <matrix> 2 matrix-p-norm-entrywise ."
+ "4.0"
+ }
} ;
HELP: matrix-p-norm
-{ $values { "m" matrix } { "p" "positive real number" } { "n" "non-negative real number" } }
-{ $description "Find the size of a matrix in ๐โ (" { $snippet "L^p" } ") vector space, usually written โฅ๏ฝฅโฅโ. For " { $snippet "p โ 1, 2, โ" } ", this is an \"entry-wise\" norm." }
+{ $values { "m" matrix } { "p" "a positive real number" } { "n" "a non-negative real number" } }
+{ $description "Find the norm (size) of a matrix in ๐โ (" { $snippet "L^p" } ") vector space, usually written โฅ๏ฝฅโฅโ. For " { $snippet "p โ 1, 2, โ" } ", this is an \"entry-wise\" norm." }
+{ $notelist
+ { "User code should call the generic " { $link p-norm } " instead." }
+ { $equiv-word-note "matrix-specific" p-norm }
+ { $equiv-word-note { $snippet "p = q" } matrix-p-q-norm }
+ $2d-only-note
+}
{ $examples
- "Calls " { $link m-1norm } ":"
- { $example
- "USING: math.matrices prettyprint ;"
- "4 4 1 <matrix> 1 matrix-p-norm ."
- "4"
- }
- "Falls back to " { $link matrix-p-norm-entrywise } ":"
- { $example
- "USING: math.functions math.matrices prettyprint ;"
- "2 2 3 <matrix> 1.5 matrix-p-norm 7.559 10e-4 ~ ."
- "t"
- }
+ "Calls " { $link l1-norm } ":"
+ { $example
+ "USING: math.matrices prettyprint ;"
+ "4 4 1 <matrix> 1 matrix-p-norm ."
+ "4"
+ }
+ "Falls back to " { $link matrix-p-norm-entrywise } ":"
+ { $example
+ "USING: math.functions math.matrices prettyprint ;"
+ "2 2 3 <matrix> 1.5 matrix-p-norm 7.559 10e-4 ~ ."
+ "t"
+ }
} ;
{ matrix-p-norm matrix-p-norm-entrywise } related-words
-{ matrix-p-norm matrix-p-q-norm } related-words
+{ matrix-l1-norm matrix-l2-norm matrix-l-infinity-norm matrix-p-norm matrix-p-q-norm } related-words
-HELP: normalize-matrix
-{ $values { "m" matrix } { "m'" matrix } }
-{ $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" } "." }
+HELP: matrix-normalize
+{ $values { "m" "a matrix with at least 1 non-zero number" } { "m'" matrix } }
+{ $description "Normalize a matrix containing at least 1 non-zero element. Each element from the input matrix is computed as a fraction of the maximum element. The maximum element becomes " { $snippet "1/1" } "." }
{ $notelist
- $2d-only-note
- { $matrix-scalar-note max abs / }
+ $2d-only-note
+ { $matrix-scalar-note max abs / }
}
{ $examples
- { $example
- "USING: math.matrices prettyprint ;"
- "{ { 5 9 } { 15 17 } } normalize-matrix ."
- "{ { 5/17 9/17 } { 15/17 1 } }"
- }
+ { $example
+ "USING: math.matrices prettyprint ;"
+ "{ { 5 9 } { 15 17 } } matrix-normalize ."
+ "{ { 5/17 9/17 } { 15/17 1 } }"
+ }
} ;
HELP: main-diagonal
projects / factor.git / blobdiff