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Article contents:[[tags: egg]] == math-utils [[toc:]] == Documentation === Miscellaneous Math Functions ==== Usage <enscript language=scheme> (import math-utils) </enscript> ==== log-with-base ==== log/base <procedure>(log-with-base B) -> (real -> real)</procedure> <procedure>(log/base B) -> (real -> real)</procedure> Returns a function for the base {{B}} logarithm. ==== coprime? <procedure>(coprime? M [N0 ...]) -> boolean</procedure> Are the integers {{M N0 ...}} coprime? ==== pairwise-coprime? <procedure>(pairwise-coprime? M [N0 ...]) -> boolean</procedure> Are the pairs of integers in {{M N0 ...}} coprime? ==== fxcoprime? <procedure>(fxcoprime? M N) -> boolean</procedure> Are the fixnums {{M N}} coprime? <enscript language=scheme> (import (only (srfi 1) filter iota)) (import (only (math-utils) fxcoprime?)) (define (coprimes n) (filter (cut fxcoprime? n <>) (iota (- n 1) 1)) ) </enscript> <enscript language=scheme> (import (only (streams derived) stream-range stream-filter)) (import (only (math-utils) fxcoprime?)) (define (coprime-numbers-stream n) (stream-filter (cut fxcoprime? n <>) (stream-range 1 n)) ) </enscript> ==== simple-interest <procedure>(simple-interest RATE TIME [PRIN]) -> number</procedure> The accumulation function, the principle is assumed {{1}}. Returns the simple interest for the {{RATE}} over {{TIME}}. ; {{RATE}} : {{number}} ; interest rate ; {{TIME}} : {{number}} ; time period to cover ; {{PRIN}} : {{number}} ; principle, default {{1}} ==== compound-interest <procedure>(compound-interest RATE FREQ TIME [PRIN]) -> number</procedure> The accumulation function, the principle is assumed {{1}}. Returns the compound interest for the {{RATE}}, applied with {{FREQ}}, over {{TIME}}. ; {{RATE}} : {{number}} ; interest rate ; {{FREQ}} : {{number}} ; compounding frequency ; {{TIME}} : {{number}} ; time period to cover ; {{PRIN}} : {{number}} ; principle, default {{1}} <enscript language=scheme> (compound-interest 0.043 4 6 1500) ;=> 1938.84 ;rounded to 2 places </enscript> ==== fibonacci ==== *fibonacci ==== fibonacci/memo ==== *fibonacci/memo <procedure>(fibonacci N) -> integer</procedure> <procedure>(*fibonacci N) -> integer</procedure> <procedure>(fibonacci/memo N) -> integer</procedure> <procedure>(*fibonacci/memo N) -> integer</procedure> Returns fibonacci of {{integer}} {{N}}. ==== fibonacci/approximate <procedure>(fibonacci/approximate N) -> real</procedure> Returns an approximate fibonacci of {{integer}} {{N}}, with an error of > 1 at N ~= 70. ==== binomial <procedure>(binomial N1 N2) -> integer</procedure> Returns the {{Binomial}} in {{N1}} to {{N2}}. ; {{N1}} : {{integer}} ; {{N2}} : {{integer}} ==== cross-ratio <procedure>(cross-ratio N1 N2 N3 N4) -> number</procedure> Returns the {{Cross-ratio}} of {{N1}}, {{N2}} and {{N3}}, {{N4}}. ; {{N1}} : {{number}} ; {{N2}} : {{number}} ; {{N3}} : {{number}} ; {{N4}} : {{number}} ==== ~= <procedure>(~= A B [EPSILON 1e-05]) -> boolean</procedure> Does {{A}} approximately equal {{B}}? ==== chinum ==== *chinum <procedure>(chinum N) -> fixnum</procedure> <syntax>(*chinum N) -> fixnum</syntax> Returns {{1}} for {{even?}} and {{-1}} for {{odd}}; {{(expt -1 N)}}. ==== square ==== sqr <procedure>(square N) -> number</procedure> <syntax>(sqr N) -> number</syntax> ==== cube ==== cub <procedure>(cube N) -> number</procedure> <syntax>(cub N) -> number</syntax> ==== *average <procedure>(*average LS) -> number</procedure> ; {{LS}} : {{(list-of number)}} ==== average <procedure>(average [N0 ...]) -> number</procedure> ; {{N0}} : {{number}} ==== average* <procedure>(average* [TREE ...]) -> number</procedure> ; {{TREE}} : {{(tree-of number)}} ==== least-squares <procedure>(least-squares PNTS) -> number number</procedure> Returns {{b}} & {{e}} such that {{y ~= b * x + e}}. ; {{PNTS}} : {{(list-of (pair number number))}} : list of x,y pairs ==== trapezoid <procedure>(trapezoid F N1 N2) -> (fixnum -> number)</procedure> Returns a function to calculate the area under {{F}} between {{N1}} & {{N2}} using the Trapezoid Rule. The function takes the number of estimations as an argument, larger for a "better" result. ; {{F}} : {{(number -> number)}} ; {{N1}} : {{number}} ; {{N2}} : {{number}} ==== euclidian-distance <procedure>(euclidian-distance X1 Y1 X2 Y2) -> number</procedure> ; {{X1 Y1}} : {{number number}} ; start point ; {{X2 Y2}} : {{number number}} ; end point ==== manhattan-distance <procedure>(manhattan-distance X1 Y1 X2 Y2) -> number</procedure> ; {{X1 Y1}} : {{number number}} ; start point ; {{X2 Y2}} : {{number number}} ; end point ==== factorial ==== *factorial ==== factorial/memo ==== *factorial/memo <procedure>(factorial N) -> integer</procedure> <procedure>(*factorial N) -> integer</procedure> <procedure>(factorial/memo N) -> integer</procedure> <procedure>(*factorial/memo N) -> integer</procedure> ; {{N}} : {{integer}} ==== factorial- ==== *factorial- <procedure>(factorial- N FALL) -> real</procedure> <procedure>(*factorial- N FALL) -> real</procedure> The falling factorial. ; {{N}} : {{real}} ; {{FALL}} : {{integer}} ==== factorial+ ==== *factorial+ <procedure>(factorial+ N RISE) -> real</procedure> <procedure>(*factorial+ N RISE) -> real</procedure> The rising factorial. ; {{N}} : {{real}} ; {{RISE}} : {{integer}} ==== harmonic ==== *harmonic ==== harmonic/memo ==== *harmonic/memo <procedure>(harmonic N) -> real</procedure> <procedure>(*harmonic N) -> real</procedure> procedure>(harmonic/memo N) -> real</procedure> <procedure>(*harmonic/memo N) -> real</procedure> Result of ''Harmonic'' series expansion to {{N}} terms. ; {{N}} : {{integer}} ==== euclidian-distance <procedure>(euclidian-distance X1 Y1 X2 Y2) -> number</procedure> Distance between {{X1,Y1}} and {{X2,Y2}}. ; {{X1 Y1}} : {{number number}} ; from point ; {{X2 Y2}} : {{number number}} ; to point ==== manhattan-distance <procedure>(manhattan-distance X1 Y1 X2 Y2) -> number</procedure> Distance between {{X1,Y1}} and {{X2,Y2}}. ; {{X1 Y1}} : {{number number}} ; from point ; {{X2 Y2}} : {{number number}} ; to point ==== to-places ==== @prec <syntax>(to-places PREC (FUNC EXPR))</syntax> <procedure>(@prec PREC EXPR [FUNC truncate])</procedure> Truncate the result of {{(FUNC EXPR)}} to {{PREC}} decimal places. ; {{PREC}} : {{fixnum}} ; precision ; {{FUNC}} : {{symbol}} ; ''clip'' operation, ''i.e.'' {{round}}, etc. ; {{EXPR}} : {{*}} ; value to ''clip'' ==== big-pi <syntax>(big-pi F N1 N2) -> number</syntax> Product of {{F}} in {{N1}} to {{N2}}. ; {{F}} : {{(number -> number)}} ; ; {{N1}} : {{number}} ; ; {{N2}} : {{number}} ; ==== big-sigma <syntax>(big-sigma F N1 N2) -> number</syntax> Sum of {{F}} in {{N1}} to {{N2}}. ; {{F}} : {{(number -> number)}} ; ; {{N1}} : {{number}} ; ; {{N2}} : {{number}} ; ==== prod <syntax>(prod VAR ([N1 [N2 [INC]]]) BODY...) -> number</syntax> Product of {{BODY...}} in {{N1}} to {{N2}} by {{INC}}. ; {{VAR}} : {{symbol}} ; {{VAR}} bound to current {{N}} in {{BODY...}} ; {{BODY...}} : {{number}} ; ; {{N1}} : {{number}} ; ; {{N2}} : {{number}} ; ; {{INC}} : {{number}} ; ==== summ <syntax>(summ VAR ([N1 [N2 [INC]]]) BODY...) -> number</syntax> Sum of {{BODY...}} in {{N1}} to {{N2}} by {{INC}}. ; {{VAR}} : {{symbol}} ; {{VAR}} bound to current {{N}} in {{BODY...}} ; {{BODY...}} : {{number}} ; ; {{N1}} : {{number}} ; ; {{N2}} : {{number}} ; ; {{INC}} : {{number}} ; === Miscellaneous Vector Math Functions ==== Usage <enscript language=scheme> (import (math-utils vector)) </enscript> ==== absolute-magnitude <procedure>(absolute-magnitude NUMVEC) -> number</procedure> ; {{NUMVEC}} : {{(vector-of number)}} : . ==== cosine-similarity <procedure>(cosine-similarity NUMVEC1 NUMVEC2) -> number</procedure> ; {{NUMVEC1}} : {{(vector-of number)}} : . ; {{NUMVEC2}} : {{(vector-of number)}} : . Must be same {{vector-length}}. ==== vector-compare ==== vector-compare2 <procedure>(vector-compare NUMVEC ...) -> number</procedure> <procedure>(vector-compare2 NUMVEC NUMVEC) -> number</procedure> Result is {{negative}}, {{zero}}, or {{positive}}. Comparison by length when unequal & element-wise when equal. ; {{NUMVEC}} : {{(vector-of number)}} : . ==== dot-product <procedure>(dot-product NUMVEC1 NUMVEC2) -> number</procedure> ; {{NUMVEC1}} : {{(vector-of number)}} : . ; {{NUMVEC2}} : {{(vector-of number)}} : . Must be same {{vector-length}}. ==== cross-product <procedure>(cross-product NUMVEC1 NUMVEC2) -> (or number (vector-of number))</procedure> Only defined for a {{vector-length}} of {{(0 1 2 3 4 8)}}. All others generate an {{error}}. Must be same {{vector-length}}. ; {{NUMVEC1}} : {{(vector-of number)}} : . ; {{NUMVEC2}} : {{(vector-of number)}} : . ==== vector-sum <procedure>(vector-sum NUMVEC) -> number</procedure> ==== softmax <procedure>(softmax NUMVEC) -> (vector-of number)</procedure> Returns the ''softmax'' of the vector elements. Normalize the vector elements so {{(= 1 (vector-sum NUMVEC))}}. ==== softmax* <procedure>(softmax* NUMVEC [TEMP]) -> (vector-of number)</procedure> Returns the ''softmax'' of the ''tempered'' vector elements. Spread, {{(< 1 TEMP)}}, or shrink, {{(> 1 TEMP)}}, data before normalization. A {{TEMP}} of {{1}} has no effect. ; {{TEMP}} : {{real}} ; ''temperature'', default {{1}}. '''Note''' that a {{TEMP}} ''near'' zero is currently an error! ==== vector-apply <procedure>(vector-apply FUNC VEC1 VEC2 [VEC...]) -> vector</procedure> ; {{FUNC}} : {{(* * !#rest * -> *)}} : . ; {{VEC#}} : {{vector}} : . Must be same {{vector-length}}. <enscript language=scheme> (import (only (math-utils vector) vector-apply)) (define (vector-mul . vs) (apply vector-apply * vs)) (define (vector-add . vs) (apply vector-apply + vs)) (define (vector-dif . vs) (apply vector-apply - vs)) </enscript> ==== vector-scale ==== vector-shift <procedure>(vector-scale VEC FACTOR) -> vector</procedure> <procedure>(vector-shift VEC BIAS) -> vector</procedure> ==== normalize-vector-scale ==== normalize-vector-shift <procedure>(normalize-vector-scale VEC) -> vector</procedure> <procedure>(normalize-vector-shift VEC) -> vector</procedure> {{vector-scale}} by the {{(/ (vector-maximum))}}. {{vector-shift}} by the {{(- (vector-minimum))}}. == Examples <enscript language=scheme> (import scheme) (import (chicken base)) (import (rename (only math-utils summ sqr ;non-checking & memoized version is used *chinum factorial/memo *factorial+/memo) (*chinum chinum) (factorial/memo factorial) (*factorial+/memo factorial+))) ;https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.132.221601 ;@lam `lambda' param (define (saha-sinha-pi #!optional (lim 100) (lam 4)) (+ 4 (summ n (1 lim) (let* ((2n (* 2 n)) (2n+1 (+ 2n 1))) (* (/ 1 (factorial n)) (- (/ 1 (+ n lam)) (/ 4 2n+1)) (factorial+ (- (/ (sqr 2n+1) (+ (* 2 2n) (* 4 lam))) n) (- n 1)))))) ) (define (madhava-leibniz-pi #!optional (lim 100)) (* 4 (summ n (1 lim) (/ (chinum (- n 1)) (- (* 2 n) 1)))) ) (- (acos -1) (saha-sinha-pi)) ;=> 1.90993887372315e-11 (- (acos -1) (madhava-leibniz-pi)) ;=> -0.0101007524813226 </enscript> == Requirements [[memoize]] [[miscmacros]] [[vector-lib]] [[test]] [[test-utils]] == Author [[/users/kon-lovett|Kon Lovett]] == Repository This egg is hosted on the CHICKEN Subversion repository: [[https://anonymous@code.call-cc.org/svn/chicken-eggs/release/5/math-utils|https://anonymous@code.call-cc.org/svn/chicken-eggs/release/5/math-utils]] If you want to check out the source code repository of this egg and you are not familiar with Subversion, see [[/egg-svn-checkout|this page]]. == Version history ; 1.6.0 : Add {{~=}}, {{vector-scale}}, {{vector-shift}}, {{normalize-vector-scale}}, {{normalize-vector-shift}}. ; 1.5.3 : {{factorial+-/memo}}. ; 1.5.2 : . ; 1.5.1 : Add {{*average}}, {{average*}}. Fix signatures. ; 1.5.0 : Add {{softmax*}}, {{chinum}}, {{*chinum}}, {{summ}}, {{prod}}, {{sqr}}, {{cub}}, {{*...}} (no type checks) & {{../memo}} (memoized) series routines. Rename {{*fibonacci}} {{fibonacci/approximate}} ; 1.4.0 : Add {{softmax}}. Allow {{real}} arguments to {{factorial+}}, {{factorial-}}, {{harmonic}}. Add {{euclidian-distance}} & {{manhattan-distance}}. ; 1.3.2 : Fix {{cross-product}} return. ; 1.3.1 : Fix {{average}} signature. ; 1.3.0 : {{cross-product}} returns {{number-vector}}. Add {{vector-compare2}}. Rename {{fibonacci*}}. Drop deprecated. ; 1.2.0 : Add {{to-places}}. Deprecate {{with-places}}. ; 1.1.0 : Add {{with-places}}. ; 1.0.6 : Remove {{mathh}} dependency since include is unreliable. ; 1.0.5 : Use {{gcd}} based algorithm. ; 1.0.4 : Add {{fxcoprime?}}. ; 1.0.3 : {{*coprime?}} too slow. ; 1.0.2 : Fix {{mathh}} dependency. ; 1.0.1 : Add {{vector-compare}}. ; 1.0.0 : From {{mathh:4.6.5}}. == License This code is in the public domain. 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