/usr/share/axiom-20170501/src/algebra/GALUTIL.spad is in axiom-source 20170501-3.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 | )abbrev package GALUTIL GaloisGroupUtilities
++ Author: Frederic Lehobey
++ Date Created: 29 June 1994
++ Date Last Updated: 30 June 1994
++ Description:
++ \spadtype{GaloisGroupUtilities} provides several useful functions.
GaloisGroupUtilities(R) : SIG == CODE where
R : Ring
N ==> NonNegativeInteger
Z ==> Integer
SIG ==> with
pascalTriangle : (N,Z) -> R
++ pascalTriangle(n,r) returns the binomial coefficient
++ \spad{C(n,r)=n!/(r! (n-r)!)}
++ and stores it in a table to prevent recomputation.
rangePascalTriangle : N -> N
++ rangePascalTriangle(n) sets the maximal number of lines which
++ are stored and returns the previous value.
rangePascalTriangle : () -> N
++ rangePascalTriangle() returns the maximal number of lines stored.
sizePascalTriangle : () -> N
++ sizePascalTriangle() returns the number of entries currently stored
++ in the table.
fillPascalTriangle : () -> Void
++ fillPascalTriangle() fills the stored table.
if R has FloatingPointSystem then
safeCeiling : R -> Z
++ safeCeiling(x) returns the integer which is greater than any integer
++ with the same floating point number representation.
safeFloor : R -> Z
++ safeFloor(x) returns the integer which is lower or equal to the
++ largest integer which has the same floating point number
++ representation.
safetyMargin : N -> N
++ safetyMargin(n) sets to n the number of low weight digits we do not
++ trust in the floating point representation and returns the previous
++ value (for use by \spadfun{safeCeiling}).
safetyMargin : () -> N
++ safetyMargin() returns the number of low weight digits we do not
++ trust in the floating point representation (used by
++ \spadfun{safeCeiling}).
CODE ==> add
if R has FloatingPointSystem then
safetymargin : N := 6
safeFloor(x:R):Z ==
if (shift := order(x)-precision()$R+safetymargin) >= 0 then
x := x+float(1,shift)
retract(floor(x))@Z
safeCeiling(x:R):Z ==
if (shift := order(x)-precision()$R+safetymargin) >= 0 then
x := x+float(1,shift)
retract(ceiling(x))@Z
safetyMargin(n:N):N ==
(safetymargin,n) := (n,safetymargin)
n
safetyMargin():N == safetymargin
pascaltriangle : FlexibleArray(R) := empty()
ncomputed : N := 3
rangepascaltriangle : N := 216
pascalTriangle(n:N, r:Z):R ==
negative? r => 0
(d := n-r) < r => pascalTriangle(n,d)
zero? r => 1$R
(r = 1) => n :: R
n > rangepascaltriangle =>
binomial(n,r)$IntegerCombinatoricFunctions(Z) :: R
n <= ncomputed =>
m := divide(n-4,2)
mq := m.quotient
pascaltriangle((mq+1)*(mq+m.remainder)+r-1)
-- compute the missing lines
for i in (ncomputed+1)..n repeat
for j in 2..(i quo 2) repeat
pascaltriangle := concat!(pascaltriangle,pascalTriangle((i-1)
:: N, j-1)+pascalTriangle((i-1) :: N,j))
ncomputed := i
pascalTriangle(n,r)
rangePascalTriangle(n:N):N ==
if n<ncomputed then
if n<3 then
pascaltriangle := delete!(pascaltriangle,1..#pascaltriangle)
ncomputed := 3
else
d := divide(n-3,2)
dq := d.quotient
pascaltriangle := delete!(pascaltriangle,((dq+1)*(dq+d.remainder)
+1)..#pascaltriangle)
ncomputed := n
(rangepascaltriangle,n) := (n,rangepascaltriangle)
n
rangePascalTriangle():N == rangepascaltriangle
sizePascalTriangle():N == #pascaltriangle
fillPascalTriangle():Void == pascalTriangle(rangepascaltriangle,2)
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