100 lines
4.7 KiB
Factor
100 lines
4.7 KiB
Factor
USING: help.markup help.syntax math sequences ;
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IN: math.polynomials
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ARTICLE: "polynomials" "Polynomials"
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"A polynomial is a vector with the highest powers on the right:"
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{ $code "{ 1 1 0 1 } -> 1 + x + x^3" "{ } -> 0" }
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"Numerous words are defined to help with polynomial arithmetic:"
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{ $subsection p= }
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{ $subsection p+ }
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{ $subsection p- }
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{ $subsection p* }
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{ $subsection p-sq }
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{ $subsection powers }
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{ $subsection n*p }
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{ $subsection p/mod }
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{ $subsection pgcd }
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{ $subsection polyval }
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{ $subsection pdiff }
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{ $subsection pextend-conv }
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{ $subsection ptrim }
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{ $subsection 2ptrim } ;
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ABOUT: "polynomials"
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HELP: powers
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{ $values { "n" integer } { "x" number } { "seq" sequence } }
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{ $description "Output a sequence having " { $snippet "n" } " elements in the format: " { $snippet "{ 1 x x^2 x^3 ... }" } "." }
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{ $examples { $example "USING: math.polynomials prettyprint ;" "4 2 powers ." "{ 1 2 4 8 }" } } ;
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HELP: p=
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{ $values { "p" "a polynomial" } { "q" "a polynomial" } { "?" "a boolean" } }
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{ $description "Tests if two polynomials are equal." }
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{ $examples { $example "USING: math.polynomials prettyprint ;" "{ 0 1 } { 0 1 0 } p= ." "t" } } ;
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HELP: ptrim
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{ $values { "p" "a polynomial" } { "p" "a polynomial" } }
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{ $description "Trims excess zeros from a polynomial." }
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{ $examples { $example "USING: math.polynomials prettyprint ;" "{ 0 1 0 0 } ptrim ." "{ 0 1 }" } } ;
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HELP: 2ptrim
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{ $values { "p" "a polynomial" } { "q" "a polynomial" } { "p" "a polynomial" } { "q" "a polynomial" } }
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{ $description "Trims excess zeros from two polynomials." }
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{ $examples { $example "USING: kernel math.polynomials prettyprint ;" "{ 0 1 0 0 } { 1 0 0 } 2ptrim [ . ] bi@" "{ 0 1 }\n{ 1 }" } } ;
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HELP: p+
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{ $values { "p" "a polynomial" } { "q" "a polynomial" } { "r" "a polynomial" } }
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{ $description "Adds " { $snippet "p" } " and " { $snippet "q" } " component-wise." }
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{ $examples { $example "USING: math.polynomials prettyprint ;" "{ 1 0 1 } { 0 1 } p+ ." "{ 1 1 1 }" } } ;
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HELP: p-
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{ $values { "p" "a polynomial" } { "q" "a polynomial" } { "r" "a polynomial" } }
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{ $description "Subtracts " { $snippet "q" } " from " { $snippet "p" } " component-wise." }
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{ $examples { $example "USING: math.polynomials prettyprint ;" "{ 1 1 1 } { 0 1 } p- ." "{ 1 0 1 }" } } ;
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HELP: n*p
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{ $values { "n" number } { "p" "a polynomial" } { "n*p" "a polynomial" } }
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{ $description "Multiplies each element of " { $snippet "p" } " by " { $snippet "n" } "." }
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{ $examples { $example "USING: math.polynomials prettyprint ;" "4 { 3 0 1 } n*p ." "{ 12 0 4 }" } } ;
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HELP: pextend-conv
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{ $values { "p" "a polynomial" } { "q" "a polynomial" } { "p" "a polynomial" } { "q" "a polynomial" } }
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{ $description "Convulution, extending to " { $snippet "p_m + q_n - 1" } "." }
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{ $examples { $example "USING: kernel math.polynomials prettyprint ;" "{ 1 0 1 } { 0 1 } pextend-conv [ . ] bi@" "V{ 1 0 1 0 }\nV{ 0 1 0 0 }" } } ;
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HELP: p*
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{ $values { "p" "a polynomial" } { "q" "a polynomial" } { "r" "a polynomial" } }
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{ $description "Multiplies two polynomials." }
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{ $examples { $example "USING: math.polynomials prettyprint ;" "{ 1 2 3 0 0 0 } { 1 2 0 0 } p* ." "{ 1 4 7 6 0 0 0 0 0 }" } } ;
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HELP: p-sq
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{ $values { "p" "a polynomial" } { "p^2" "a polynomial" } }
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{ $description "Squares a polynomial." }
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{ $examples { $example "USING: math.polynomials prettyprint ;" "{ 1 2 0 } p-sq ." "{ 1 4 4 0 0 }" } } ;
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HELP: p/mod
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{ $values { "p" "a polynomial" } { "q" "a polynomial" } { "z" "a polynomial" } { "w" "a polynomial" } }
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{ $description "Computes to quotient " { $snippet "z" } " and remainder " { $snippet "w" } " of dividing " { $snippet "p" } " by " { $snippet "q" } "." }
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{ $examples { $example "USING: kernel math.polynomials prettyprint ;" "{ 1 1 1 1 } { 3 1 } p/mod [ . ] bi@" "V{ 7 -2 1 }\nV{ -20 0 0 }" } } ;
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HELP: pgcd
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{ $values { "p" "a polynomial" } { "q" "a polynomial" } { "a" "a polynomial" } { "d" "a polynomial" } }
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{ $description "Computes the greatest common divisor " { $snippet "d" } " of " { $snippet "p" } " and " { $snippet "q" } ", and another value " { $snippet "a" } " satisfying:" { $code "a*q = d mod p" } }
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{ $notes "GCD in the case of polynomials is a monic polynomial of the highest possible degree that divides into both " { $snippet "p" } " and " { $snippet "q" } "." }
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{ $examples
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{ $example "USING: kernel math.polynomials prettyprint ;"
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"{ 1 1 1 1 } { 1 1 } pgcd [ . ] bi@"
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"{ 0 0 }\n{ 1 1 }"
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}
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} ;
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HELP: pdiff
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{ $values { "p" "a polynomial" } { "p'" "a polynomial" } }
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{ $description "Finds the derivative of " { $snippet "p" } "." } ;
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HELP: polyval
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{ $values { "p" "a polynomial" } { "x" number } { "p[x]" number } }
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{ $description "Evaluate " { $snippet "p" } " with the input " { $snippet "x" } "." }
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{ $examples { $example "USING: math.polynomials prettyprint ;" "{ 1 0 1 } 2 polyval ." "5" } } ;
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