% -*- Mode: Postscript -*- % pst-math.pro --- PostScript header file pst-math.pro % % Author : Christophe JORSSEN % Author : Herbert Voß % Created the : Sat 20 March 2004 % Last Mod : $Date: 2009/08/27 $ % Version : 0.5 $ % /PI 3.14159265359 def /ENeperian 2.71828182846 def % /DegToRad {PI mul 180 div} bind def /RadToDeg {180 mul PI div} bind def % /COS {RadToDeg cos} bind def /SIN {RadToDeg sin} bind def /TAN {dup SIN exch COS Div} bind def /tan {dup sin exch cos Div} bind def /ATAN {neg -1 atan 180 sub DegToRad} bind def /ACOS {dup dup mul neg 1 add sqrt exch atan DegToRad} bind def /acos {dup dup mul neg 1 add sqrt exch atan} bind def /ASIN {neg dup dup mul neg 1 add sqrt neg atan 180 sub DegToRad} bind def /asin {neg dup dup mul neg 1 add sqrt neg atan 180 sub} bind def % /EXP {ENeperian exch exp} bind def % /COSH {dup EXP exch neg EXP add 2 div} bind def /SINH {dup EXP exch neg EXP sub 2 div} bind def /TANH {dup SINH exch COSH div} bind def /ACOSH {dup dup mul 1 sub sqrt add ln} bind def /ASINH {dup dup mul 1 add sqrt add ln} bind def /ATANH {dup 1 add exch neg 1 add Div ln 2 div} bind def % %/SINC {dup SIN exch Div} bind def /SINC { dup 0 eq { pop 1 } { dup SIN exch div } ifelse } bind def /GAUSS {dup mul 2 mul dup 4 -2 roll sub dup mul exch div neg EXP exch PI mul sqrt div} bind def % /GAMMA { 2 dict begin % hv 2007-08-30 /z exch def 1.000000000190015 % p(0) 0 1 5 { % on stack is 0 1 2 3 4 5 dup % n-1 n-1 [ 76.18009172947146 -86.50532032941677 24.0140982483091 -1.231739572450155 0.1208650973866179E-2 -0.5395239384953E-5 ] exch get exch % p(n) n-1 1 add z add div % p(n)/(z+n) add % build the sum } for Pi 2 mul sqrt z div mul z 5.5 add z 0.5 add exp mul ENeperian z 5.5 add neg exp mul end } bind def % /GAMMALN {dup dup dup 5.5 add dup ln 3 -1 roll .5 add mul sub neg 1.000000000190015 0 1 5 { [76.18009172947146 -86.50532032941677 24.0140982483091 -1.231739572450155 .1208650973866179E-2 -.5395239384953E-5 2.5066282746310005] exch get 4 -1 roll 1 add dup 5 1 roll div add} for 4 -1 roll div 2.5066282746310005 mul ln add exch pop} bind def /BETA {2 copy add GAMMALN neg exch GAMMALN 3 -1 roll GAMMALN EXP} bind def % /HORNER {aload length dup 2 add -1 roll exch 1 sub { dup 4 1 roll mul add exch } repeat pop } bind def % /BESSEL_J0 {dup abs 8 lt { dup mul dup [57568490574 -13362590354 651619640.7 -11214424.18 77392.33017 -184.9052456] HORNER exch [57568490411 1029532985 9494680.718 59272.64853 267.8532712 1] HORNER Div} {abs dup .636619772 exch div sqrt exch dup .785398164 sub exch 8 exch div dup dup mul dup [1 -1.098628627E-2 .2734510407E-4 -.2073370639E-5 .2093887211E-6] HORNER 3 index COS mul exch [-.1562499995E-1 .1430488765E-3 -.6911147651E-5 .7621095161E-6 -.934945152E-7] HORNER 4 -1 roll SIN mul 3 -1 roll mul neg add mul} ifelse} bind def % /BESSEL_Y0 {dup 8 lt { dup dup mul dup [-2957821389 7062834065 -512359803.6 10879881.29 -86327.92757 228.4622733] HORNER exch [40076544269 745249964.8 7189466.438 47447.26470 226.1030244 1] HORNER Div exch dup ln exch BESSEL_J0 .636619772 mul mul add} {dup .636619772 exch div sqrt exch dup .785398164 sub exch 8 exch div dup dup mul dup [1 -.1098628627E-2 .2734510407E-4 -.2073370639E-5 .2093887211E-6] HORNER 3 index SIN mul exch [-.1562499995E-1 .1430488765E-3 -.6911147651E-5 .7621095161E-6 -.934945152E-7] HORNER 4 -1 roll COS mul 3 -1 roll mul add mul} ifelse} bind def % /BESSEL_J1 {dup abs 8 lt { dup dup mul dup 3 -2 roll [72362614232 -7895059235 242396853.1 -2972611.439 15704.48260 -30.16036606] HORNER mul exch [144725228442 2300535178 18583304.74 99447.43394 376.9991397 1] HORNER Div} {dup abs dup .636619772 exch div sqrt exch dup 2.356194491 sub exch 8 exch div dup dup mul dup [1 .183105E-2 -.3516396496E-4 .2457520174E-5 -.240337019E-6] HORNER 3 index COS mul exch [.04687499995 6.2002690873E-3 .8449199096E-5 -.88228987E-6 .105787412E-6] HORNER 4 -1 roll SIN mul 3 -1 roll mul neg add mul exch dup abs Div mul} ifelse} bind def % /BESSEL_Y1 {dup 8 lt { dup dup dup mul dup [-.4900604943E13 .1275274390E13 -.5153428139E11 .7349264551E9 -.4237922726E7 .8511937935E4] HORNER exch [.2499580570E14 .4244419664E12 .3733650367E10 .2245904002E8 .1020426050E6 .3549632885E3 1] HORNER Div mul exch dup dup ln exch BESSEL_J1 mul exch 1 exch div sub .636619772 mul add} {dup .636619772 exch div sqrt exch dup 2.356194491 sub exch 8 exch div dup dup mul dup [1 .183105E-2 -.3516396496E-4 .2457520174E-5 -.240337019E-6] HORNER 3 index SIN mul exch [.04687499995 -.2002690873E-3 .8449199096E-5 6.88228987E-6 .105787412E-6] HORNER 4 -1 roll COS mul 3 -1 roll mul add mul} ifelse} bind def % % En cours... /BESSEL_Yn {dup 0 eq {pop BESSEL_Y0}{dup 1 eq {pop BESSEL_Y1}{ exch dup BESSEL_Y0 exch dup BESSEL_Y1 exch 2 exch Div { mul 3 -1 roll mul 2 index sub pstack} for } ifelse } ifelse } bind def % /SIMPSON { 1 dict begin %% on stack a b var f ierr Dominik Rodriguez 3 index 5 index sub % compute h 1 % a b var f ierr h n 4 index 7 index def 3 index exec % a b var f ierr h n f(a) 5 index 7 index def 4 index exec add % a b var f ierr h n f(a)+f(b) 5 index 8 index 4 index 2 div add def 4 index exec % a b var f ierr h n f(a)+f(b) f(a+h/2) exch 1 index 4 mul add 0 % a b var f ierr h n old=f(a+h/2) Estim=f(a)+f(b)+4f(a+h/2) NbLoop { % a b var f ierr h n old Estim NbLoop 5 -1 roll 2 div dup 6 1 roll % h<-h/2 5 -1 roll 2 mul 5 1 roll % n<-2n % a b var f ierr h n old Estim NbLoop h 2 div 10 index add 0 % a b var f ierr h n old Estim NbLoop a+h/2 Cumul 5 index { 1 index 10 index exch def 8 index exec add exch 6 index add exch } repeat % a b var f ierr h n old Estim NbLoop a+nh/2 Cumul exch pop % a b var f ierr h n old Estim NbLoop New 2 index 1 index 4 mul 6 -1 roll 2 mul sub sub % a b var f ierr h n Estim NbLoop New Diff 4 -1 roll 2 mul 1 index sub 4 1 roll % a b var f ierr h n Estim NbLoop New Diff exch 4 1 roll % a b var f ierr h n old Estim NbLoop Diff 5 index 6 div mul abs 6 index lt { exit } if 1 add dup 9 eq { exit } if } loop % a b var f ierr h n old Estim NbLoop exch 5 -1 roll 6 div mul mark 10 2 roll cleartomark end } def % ------------------------------------ math stuff ---------------------------------- % % Matrix A in arrays of rows A[[row1][row2]...] % with [row1]=[a11 a12 ... b1] % returns on stack solution vector X=[x1 x2 ... xn] /SolveLinEqSystem { % on stack matrix M=[A,b] (A*x=b) 10 dict begin % hold all ocal /A exch def /Rows A length def % Rows = number of rows /Cols A 0 get length def % Cols = number of columns /Index [ 0 1 Rows 1 sub { } for ] def % Index = [0 1 2 ... Rows-1] /col 0 def /row 0 def /PR Rows array def % PR[c] = pivot row for row row { % starts the loop, find pivot entry in row r col Cols ge row Rows ge or { exit } if % col < Cols and row < Rows else exit /pRow row def % pRow = pivot row /max A row get col get abs def % get A[row[col]], first A[0,0] row 1 add 1 Rows 1 sub { % starts for loop 1 1 Rows-1 /j exch def % index counter /x A j get col get abs def % get A[j[r]] x max gt { % x>max, then save position /pRow j def /max x def } if } for % now we have the row with biggest A[0,1] % with pRow = the pivot row max 0 gt { % swap entries pRow and row in i /tmp Index row get def Index row Index pRow get put Index pRow tmp put % and columns pRow and row in A /tmp A row get def A row A pRow get put A pRow tmp put % pivot /row0 A row get def % the pivoting row /p0 row0 col get def % the pivot value row 1 add 1 Rows 1 sub { % start for loop /j exch def /c1 A j get def /p c1 col get p0 div def c1 col p put % subtract (p1/p0)*row[i] from row[j] col 1 add 1 Cols 1 sub { % start for loop /i exch def c1 dup i exch % c1 i c1 i get row0 i get p mul sub put } for } for PR row col put /col col 1 add def /row row 1 add def }{ % all zero entries /row row 1 add def % continue loop with same row } ifelse } loop /X A def % solution vector A Rows 1 sub get dup Cols 1 sub get exch Cols 2 sub get div X Rows 1 sub 3 -1 roll put % X[n] Rows 2 sub -1 0 { % for loop to calculate X[i] /xi exch def % current index A xi get % i-th row /Axi exch def /sum 0 def Cols 2 sub -1 xi 1 add { /n exch def /sum sum Axi n get X n get mul add def } for Axi Cols 1 sub get % b=Axi[Cols-1] sum sub % b-sum Axi xi get div % b-sum / Axi[xi] X xi 3 -1 roll put % X[xi] } for X end } def % % % END pst-math.pro