% -*- Mode: Postscript -*-
% pst-math.pro --- PostScript header file pst-math.pro
%
% Author : Christophe JORSSEN
% Author : Herbert Voß <hvoss@tug.org>
% 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