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[Axiom-developer] 20080822.01.tpd.patch (add linalg, overload regression
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20080822.01.tpd.patch (add linalg, overload regressions) |
|
Date: |
Fri, 22 Aug 2008 01:49:12 -0500 |
linalg.input and overload.input are two new regression test files
=================================================================
diff --git a/changelog b/changelog
index 3ba51a4..965fce8 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,6 @@
+20080822 tpd src/input/Makefile add linalg, overload regression tests
+20080822 tpd src/input/linalg.input recovered
+20080822 tpd src/input/overload.input recovered
20080821 tpd src/algebra/Makefile add MappingPackage4
20080821 tpd src/algebra/exposed.lsp add MappingPackage4
20080821 tpd src/algebra/mappkg.spad add MAPPKG4
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet
index 0d919d6..3f80d67 100644
--- a/src/input/Makefile.pamphlet
+++ b/src/input/Makefile.pamphlet
@@ -244,9 +244,6 @@ WRAPPED=loop.output
# Error: Value stack overflow.
VALUESTACK=tutchap67.output
-# Broken at |MATCAT-;*;3S;29|. Type :H for Help.
-MATCAT=linalg.output
-
# Broken at |STAGG-;ELT;AIS;5|. Type :H for Help.
STAGG=reductio.output
@@ -326,7 +323,8 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress
alist.regress \
kamke0.regress kamke1.regress kamke2.regress kamke3.regress \
kamke4.regress kamke5.regress kamke6.regress kamke7.regress \
kernel.regress knot2.regress kovacic.regress kuipers.regress \
- laplace.regress leg.regress limit.regress lindep.regress \
+ laplace.regress leg.regress limit.regress linalg.regress \
+ lindep.regress \
list.regress lode.regress lodesys.regress lodo1.regress \
lodo2.regress lodo3.regress lodof.regress lodo.regress \
log.regress \
@@ -341,6 +339,7 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress
alist.regress \
nqip.regress nsfip.regress numbers.regress octonion.regress \
oct.regress ode.regress odpol.regress op1.regress \
opalg.regress operator.regress op.regress ovar.regress \
+ overload.regress \
padic.regress parabola.regress pascal1.regress pascal.regress \
patch51.regress page.regress \
patmatch.regress pat.regress perman.regress perm.regress \
@@ -594,6 +593,7 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input
${OUT}/algfacob.input \
${OUT}/kovacic.input ${OUT}/kuipers.input \
${OUT}/laplace.input ${OUT}/leg.input ${OUT}/lexp.input \
${OUT}/lextripk.input ${OUT}/lib.input ${OUT}/limit.input \
+ ${OUT}/linalg.input \
${OUT}/lindep.input ${OUT}/liss1.input ${OUT}/liss2.input \
${OUT}/list.input ${OUT}/lode.input ${OUT}/lodesys.input \
${OUT}/lodo1.input ${OUT}/lodo2.input ${OUT}/lodof.input \
@@ -613,10 +613,10 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input
${OUT}/algfacob.input \
${OUT}/nepip.input ${OUT}/newton.input \
${OUT}/nlode.input ${OUT}/none.input ${OUT}/noonburg.input \
${OUT}/noptip.input ${OUT}/nqip.input ${OUT}/nsfip.input \
- ${OUT}/ntube.input ${OUT}/oct.input ${OUT}/ode.input \
+ ${OUT}/ntube.input ${OUT}/oct.input ${OUT}/ode.input \
${OUT}/octonion.input ${OUT}/odpol.input \
${OUT}/opalg.input ${OUT}/operator.input ${OUT}/op.input \
- ${OUT}/op1.input ${OUT}/ovar.input \
+ ${OUT}/op1.input ${OUT}/ovar.input ${OUT}/overload.input \
${OUT}/padic.input ${OUT}/palette.input \
${OUT}/parpcurv.input ${OUT}/parscurv.input ${OUT}/parsurf.input \
${OUT}/pascal1.input \
@@ -927,6 +927,7 @@ DOCFILES= \
${DOC}/odpol.input.dvi ${DOC}/op1.input.dvi \
${DOC}/opalg.input.dvi ${DOC}/operator.input.dvi \
${DOC}/op.input.dvi ${DOC}/ovar.input.dvi \
+ ${DOC}/overload.input.dvi \
${DOC}/padic.input.dvi ${DOC}/palette.input.dvi \
${DOC}/parabola.input.dvi ${DOC}/parpcurv.input.dvi \
${DOC}/parscurv.input.dvi ${DOC}/parsurf.input.dvi \
diff --git a/src/input/linalg.input.pamphlet b/src/input/linalg.input.pamphlet
index c39e5c7..17156fe 100644
--- a/src/input/linalg.input.pamphlet
+++ b/src/input/linalg.input.pamphlet
@@ -10,104 +10,929 @@
\tableofcontents
\eject
<<*>>=
+)spool linalg.output
+)set message test on
+)set message auto off
+)set break resume
+)clear all
-- Input for page MatrixMoreFunctionsPage
)clear all
+--S 1 of 82
m1 := matrix([[1,-2,1],[4,2,-4]])
+--R
+--R +1 - 2 1 +
+--R (1) | |
+--R +4 2 - 4+
+--R Type: Matrix
Integer
+--E 1
+
+--S 2 of 82
m2 := matrix([[0,1,2],[2,3,4],[3,4,5]])
+--R
+--R +0 1 2+
+--R | |
+--R (2) |2 3 4|
+--R | |
+--R +3 4 5+
+--R Type: Matrix
Integer
+--E 2
+
+--S 3 of 82
m3 := matrix([[1,2,3],[2,4,6]])
+--R
+--R +1 2 3+
+--R (3) | |
+--R +2 4 6+
+--R Type: Matrix
Integer
+--E 3
+
+--S 4 of 82
m1 + m3
+--R
+--R +2 0 4+
+--R (4) | |
+--R +6 6 2+
+--R Type: Matrix
Integer
+--E 4
+
+--S 5 of 82
100 * m1
+--R
+--R +100 - 200 100 +
+--R (5) | |
+--R +400 200 - 400+
+--R Type: Matrix
Integer
+--E 5
+
+--S 6 of 82
m1 * m2
+--R
+--R +- 1 - 1 - 1+
+--R (6) | |
+--R +- 8 - 6 - 4+
+--R Type: Matrix
Integer
+--E 6
+
+--S 7 of 82
-m1 + m3 * m2
+--R
+--R +12 21 24+
+--R (7) | |
+--R +22 36 54+
+--R Type: Matrix
Integer
+--E 7
+
+--S 8 of 82
m2 * m1
+--R
+--R
+--RDaly Bug
+--R >> Error detected within library code:
+--R can't multiply matrices of incompatible dimensions
+--R
+--R Continuing to read the file...
+--R
+--E 8
+
+--S 9 of 82
v := vector([1,0,1])
+--R
+--R (8) [1,0,1]
+--R Type: Vector
NonNegativeInteger
+--E 9
+
+--S 10 of 82
m3 * v
+--R
+--R (9) [4,8]
+--R Type: Vector
Integer
+--E 10
+
+--S 11 of 82
m5 : MATRIX POLY INT := new(4,4,1)
+--R
+--R +1 1 1 1+
+--R | |
+--R |1 1 1 1|
+--R (10) | |
+--R |1 1 1 1|
+--R | |
+--R +1 1 1 1+
+--R Type: Matrix Polynomial
Integer
+--E 11
+
+--S 12 of 82
vars : LIST POLY INT := [x,y,z,u]
+--R
+--R (11) [x,y,z,u]
+--R Type: List Polynomial
Integer
+--E 12
+
+--S 13 of 82
for i in 1..4 repeat for j in 1..3 repeat m5(i,j + 1) := (vars.i)**j
+--R Type:
Void
+--E 13
+
+--S 14 of 82
m5
+--R
+--R + 2 3+
+--R |1 x x x |
+--R | |
+--R | 2 3|
+--R |1 y y y |
+--R (13) | |
+--R | 2 3|
+--R |1 z z z |
+--R | |
+--R | 2 3|
+--R +1 u u u +
+--R Type: Matrix Polynomial
Integer
+--E 14
+
+--S 15 of 82
trace(m5)
+--R
+--R 2 3
+--R (14) z + y + u + 1
+--R Type: Polynomial
Integer
+--E 15
+
+--S 16 of 82
det := determinant(m5)
+--R
+--R (15)
+--R 2 2 2 2 2 3
+--R ((- x + u)y + (x - u )y - u x + u x)z
+--R +
+--R 3 3 3 3 3 2
+--R ((x - u)y + (- x + u )y + u x - u x)z
+--R +
+--R 2 2 3 3 3 2 2 3 3 2 2 2 3
+--R ((- x + u )y + (x - u )y - u x + u x )z + (u x - u x)y
+--R +
+--R 3 3 2 2 3 3 2
+--R (- u x + u x)y + (u x - u x )y
+--R Type: Polynomial
Integer
+--E 16
+
+--S 17 of 82
factor(det)
+--R
+--R (16) - (x - u)(y - x)(y - u)(z - y)(z - x)(z - u)
+--R Type: Factored Polynomial
Integer
+--E 17
+
+--S 18 of 82
m6 := matrix([[1,2,1],[-2,3,4],[-1,5,6]])
+--R
+--R + 1 2 1+
+--R | |
+--R (17) |- 2 3 4|
+--R | |
+--R +- 1 5 6+
+--R Type: Matrix
Integer
+--E 18
+
+--S 19 of 82
m6inv := inverse(m6)
+--R
+--R + 2 5 +
+--R |- - - 1 - |
+--R | 7 7 |
+--R | |
+--R (18) | 8 6|
+--R | - 1 - -|
+--R | 7 7|
+--R | |
+--R +- 1 - 1 1 +
+--R Type: Union(Matrix Fraction
Integer,...)
+--E 19
+
+--S 20 of 82
m6 * m6inv
+--R
+--R +1 0 0+
+--R | |
+--R (19) |0 1 0|
+--R | |
+--R +0 0 1+
+--R Type: Matrix Fraction
Integer
+--E 20
+
+--S 21 of 82
m7 := matrix([[1,2,1],[-2,3,4],[-1,5,5]])
+--R
+--R + 1 2 1+
+--R | |
+--R (20) |- 2 3 4|
+--R | |
+--R +- 1 5 5+
+--R Type: Matrix
Integer
+--E 21
+
+--S 22 of 82
inverse(m7)
+--R
+--R (21) "failed"
+--R Type:
Union("failed",...)
+--E 22
+
+--S 23 of 82
determinant(m7)
+--R
+--R (22) 0
+--R Type:
NonNegativeInteger
+--E 23
+
+--S 24 of 82
m8 : SQMATRIX(2,INT) := matrix([[1,2],[2,3]])
+--R
+--R +1 2+
+--R (23) | |
+--R +2 3+
+--R Type:
SquareMatrix(2,Integer)
+--E 24
+
+--S 25 of 82
m9 : SQMATRIX(2,INT) := matrix([[1,1],[0,1]])
+--R
+--R +1 1+
+--R (24) | |
+--R +0 1+
+--R Type:
SquareMatrix(2,Integer)
+--E 25
+
+--S 26 of 82
m8 ** 2
+--R
+--R +5 8 +
+--R (25) | |
+--R +8 13+
+--R Type:
SquareMatrix(2,Integer)
+--E 26
+
+--S 27 of 82
m9 ** 3
+--R
+--R +1 3+
+--R (26) | |
+--R +0 1+
+--R Type:
SquareMatrix(2,Integer)
+--E 27
+
+--S 28 of 82
mm : SQMATRIX(2,SQMATRIX(2,INT)) := matrix([[1,m8],[m9,0]])
+--R
+--R ++1 0+ +1 2++
+--R || | | ||
+--R |+0 1+ +2 3+|
+--R (27) | |
+--R |+1 1+ +0 0+|
+--R || | | ||
+--R ++0 1+ +0 0++
+--R Type:
SquareMatrix(2,SquareMatrix(2,Integer))
+--E 28
+
+--S 29 of 82
100 * mm
+--R
+--R ++100 0 + +100 200++
+--R || | | ||
+--R |+ 0 100+ +200 300+|
+--R (28) | |
+--R |+100 100+ +0 0+ |
+--R || | | | |
+--R ++ 0 100+ +0 0+ +
+--R Type:
SquareMatrix(2,SquareMatrix(2,Integer))
+--E 29
+
+--S 30 of 82
m8 * mm
+--R
+--R ++1 2+ +5 8 ++
+--R || | | ||
+--R |+2 3+ +8 13+|
+--R (29) | |
+--R |+1 3+ +0 0+ |
+--R || | | | |
+--R ++2 5+ +0 0+ +
+--R Type:
SquareMatrix(2,SquareMatrix(2,Integer))
+--E 30
+
+--S 31 of 82
mm * mm
+--R
+--R ++2 3+ +1 2++
+--R || | | ||
+--R |+2 6+ +2 3+|
+--R (30) | |
+--R |+1 1+ +3 5+|
+--R || | | ||
+--R ++0 1+ +2 3++
+--R Type:
SquareMatrix(2,SquareMatrix(2,Integer))
+--E 31
+
+--S 32 of 82
p : POLY SQMATRIX(2,INT) := m8 * x**2 + m9 * x + m8 * m9
+--R
+--R +1 2+ 2 +1 1+ +1 3+
+--R (31) | |x + | |x + | |
+--R +2 3+ +0 1+ +2 5+
+--R Type: Polynomial
SquareMatrix(2,Integer)
+--E 32
+
+--S 33 of 82
100 * p
+--R
+--R +100 200+ 2 +100 100+ +100 300+
+--R (32) | |x + | |x + | |
+--R +200 300+ + 0 100+ +200 500+
+--R Type: Polynomial
SquareMatrix(2,Integer)
+--E 33
+
+--S 34 of 82
m8 * p
+--R
+--R +5 8 + 2 +1 3+ +5 13+
+--R (33) | |x + | |x + | |
+--R +8 13+ +2 5+ +8 21+
+--R Type: Polynomial
SquareMatrix(2,Integer)
+--E 34
+
+--S 35 of 82
p * p
+--R
+--R +5 8 + 4 +4 8+ 3 +13 26+ 2 +4 12+ +7 18+
+--R (34) | |x + | |x + | |x + | |x + | |
+--R +8 13+ +4 8+ +20 41+ +4 12+ +12 31+
+--R Type: Polynomial
SquareMatrix(2,Integer)
+--E 35
-- Input for page MatrixCanonicalFormsPage
)clear all
+--S 36 of 82
m1 := matrix([[0,4,1],[5,3,-7],[-5,5,9]])
+--R
+--R + 0 4 1 +
+--R | |
+--R (1) | 5 3 - 7|
+--R | |
+--R +- 5 5 9 +
+--R Type: Matrix
Integer
+--E 36
+
+--S 37 of 82
rank(m1)
+--R
+--R (2) 2
+--R Type:
PositiveInteger
+--E 37
+
+--S 38 of 82
rowEchelon(m1)
+--R
+--R +5 3 - 7+
+--R | |
+--R (3) |0 4 1 |
+--R | |
+--R +0 0 0 +
+--R Type: Matrix
Integer
+--E 38
+
+--S 39 of 82
nullSpace(m1)
+--R
+--R (4) [[31,- 5,20]]
+--R Type: List Vector
Integer
+--E 39
+
+--S 40 of 82
t := eigenMatrix(m1)
+--R
+--R + +----+ +----+ +
+--R |\|- 11 + 2 - \|- 11 + 2 31 |
+--R |----------- ------------- -- |
+--R | 5 5 20 |
+--R | |
+--R (5) | +----+ +----+ |
+--R |2\|- 11 - 1 - 2\|- 11 - 1 1|
+--R |------------ -------------- - -|
+--R | 5 5 4|
+--R | |
+--R + 1 1 1 +
+--R Type: Union(Matrix Expression
Integer,...)
+--E 40
+
+--S 41 of 82
inverse(t) * m1 * t
+--R
+--R + +----+ +
+--R |5581634906\|- 11 - 55255461173 |
+--R |------------------------------- 0 0|
+--R | +----+ |
+--R | 1888197247\|- 11 - 5747548576 |
+--R | |
+--R (6) | +----+ |
+--R | 6\|- 11 + 11 |
+--R | 0 ------------- 0|
+--R | +----+ |
+--R | \|- 11 |
+--R | |
+--R + 0 0 0+
+--R Type: Matrix Expression
Integer
+--E 41
-- Input for page MatrixBasicFunctionsPage
)clear all
+--S 42 of 82
m1 := matrix([[1,2,3,4],[2,3,4,5],[3,4,5,6],[4,5,6,7]])
+--R
+--R +1 2 3 4+
+--R | |
+--R |2 3 4 5|
+--R (1) | |
+--R |3 4 5 6|
+--R | |
+--R +4 5 6 7+
+--R Type: Matrix
Integer
+--E 42
+
+--S 43 of 82
m2 := matrix([[1,0,2],[20,30,10],[0,200,100]])
+--R
+--R +1 0 2 +
+--R | |
+--R (2) |20 30 10 |
+--R | |
+--R +0 200 100+
+--R Type: Matrix
Integer
+--E 43
+
+--S 44 of 82
(m3,m4) : MATRIX PF 7
+--R Type:
Void
+--E 44
+
+--S 45 of 82
m3 := matrix([[1,0,1],[5,0,1]])
+--R
+--R +1 0 1+
+--R (4) | |
+--R +5 0 1+
+--R Type: Matrix PrimeField
7
+--E 45
+
+--S 46 of 82
m4 := matrix([[1],[2],[5],[6]])
+--R
+--R +1+
+--R | |
+--R |2|
+--R (5) | |
+--R |5|
+--R | |
+--R +6+
+--R Type: Matrix PrimeField
7
+--E 46
+
+--S 47 of 82
m2(1,1)
+--R
+--R (6) 1
+--R Type:
PositiveInteger
+--E 47
+
+--S 48 of 82
m2(1,1) := 99
+--R
+--R (7) 99
+--R Type:
PositiveInteger
+--E 48
+
+--S 49 of 82
m2
+--R
+--R +99 0 2 +
+--R | |
+--R (8) |20 30 10 |
+--R | |
+--R +0 200 100+
+--R Type: Matrix
Integer
+--E 49
+
+--S 50 of 82
row(m2,2)
+--R
+--R (9) [20,30,10]
+--R Type: Vector
Integer
+--E 50
+
+--S 51 of 82
setRow!(m2,2,vector [66,77,88])
+--R
+--R +99 0 2 +
+--R | |
+--R (10) |66 77 88 |
+--R | |
+--R +0 200 100+
+--R Type: Matrix
Integer
+--E 51
+
+--S 52 of 82
r := column(m2,1)
+--R
+--R (11) [99,66,0]
+--R Type: Vector
Integer
+--E 52
+
+--S 53 of 82
setColumn!(m2,2,r)
+--R
+--R +99 99 2 +
+--R | |
+--R (12) |66 66 88 |
+--R | |
+--R +0 0 100+
+--R Type: Matrix
Integer
+--E 53
+
+--S 54 of 82
nrows(m1)
+--R
+--R (13) 4
+--R Type:
PositiveInteger
+--E 54
+
+--S 55 of 82
m5 : MATRIX INT := new(12,12,0)
+--R
+--R +0 0 0 0 0 0 0 0 0 0 0 0+
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0|
+--R (14) | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 0|
+--R | |
+--R +0 0 0 0 0 0 0 0 0 0 0 0+
+--R Type: Matrix
Integer
+--E 55
+
+--S 56 of 82
for i in 2..nrows(m5) repeat m5(i-1,i):= 1
+--R Type:
Void
+--E 56
+
+--S 57 of 82
m5
+--R
+--R +0 1 0 0 0 0 0 0 0 0 0 0+
+--R | |
+--R |0 0 1 0 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 1 0 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 1 0 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 1 0 0 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 1 0 0 0 0 0|
+--R (16) | |
+--R |0 0 0 0 0 0 0 1 0 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 1 0 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 1 0 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 1 0|
+--R | |
+--R |0 0 0 0 0 0 0 0 0 0 0 1|
+--R | |
+--R +0 0 0 0 0 0 0 0 0 0 0 0+
+--R Type: Matrix
Integer
+--E 57
+
+--S 58 of 82
d : MATRIX INT := diagonalMatrix([1,2,3,2,1])
+--R
+--R +1 0 0 0 0+
+--R | |
+--R |0 2 0 0 0|
+--R | |
+--R (17) |0 0 3 0 0|
+--R | |
+--R |0 0 0 2 0|
+--R | |
+--R +0 0 0 0 1+
+--R Type: Matrix
Integer
+--E 58
+
+--S 59 of 82
m6 := matrix([[0,1,2,3,4],[5,6,7,8,9],[10,11,12,13,14]])
+--R
+--R +0 1 2 3 4 +
+--R | |
+--R (18) |5 6 7 8 9 |
+--R | |
+--R +10 11 12 13 14+
+--R Type: Matrix
Integer
+--E 59
+
+--S 60 of 82
m7 := subMatrix(m6,1,3,2,4)
+--R
+--R +1 2 3 +
+--R | |
+--R (19) |6 7 8 |
+--R | |
+--R +11 12 13+
+--R Type: Matrix
Integer
+--E 60
+
+--S 61 of 82
horizConcat(m6,m7)
+--R
+--R +0 1 2 3 4 1 2 3 +
+--R | |
+--R (20) |5 6 7 8 9 6 7 8 |
+--R | |
+--R +10 11 12 13 14 11 12 13+
+--R Type: Matrix
Integer
+--E 61
+
+--S 62 of 82
vertConcat(m6,subMatrix(m6,1,1,1,5))
+--R
+--R +0 1 2 3 4 +
+--R | |
+--R |5 6 7 8 9 |
+--R (21) | |
+--R |10 11 12 13 14|
+--R | |
+--R +0 1 2 3 4 +
+--R Type: Matrix
Integer
+--E 62
+
+--S 63 of 82
transpose(m6)
-setsubMatrix(m6,1,3,1,3,transpose(subMatrix(m6,1,3,1,3)))
+--R
+--R +0 5 10+
+--R | |
+--R |1 6 11|
+--R | |
+--R (22) |2 7 12|
+--R | |
+--R |3 8 13|
+--R | |
+--R +4 9 14+
+--R Type: Matrix
Integer
+--E 63
+
+--S 64 of 82
+setsubMatrix!(m6,1,3,transpose(subMatrix(m6,1,3,1,3)))
+--R
+--R +0 1 0 5 10+
+--R | |
+--R (23) |5 6 1 6 11|
+--R | |
+--R +10 11 2 7 12+
+--R Type: Matrix
Integer
+--E 64
+
+--S 65 of 82
m6
+--R
+--R +0 1 0 5 10+
+--R | |
+--R (24) |5 6 1 6 11|
+--R | |
+--R +10 11 2 7 12+
+--R Type: Matrix
Integer
+--E 65
+
+--S 66 of 82
m8 := matrix([[1,2],[3,4]])
+--R
+--R +1 2+
+--R (25) | |
+--R +3 4+
+--R Type: Matrix
Integer
+--E 66
+
+--S 67 of 82
m9 := m8
+--R
+--R +1 2+
+--R (26) | |
+--R +3 4+
+--R Type: Matrix
Integer
+--E 67
+
+--S 68 of 82
m10 := copy(m8)
+--R
+--R +1 2+
+--R (27) | |
+--R +3 4+
+--R Type: Matrix
Integer
+--E 68
+
+--S 69 of 82
m8(1,1) := 1000000
+--R
+--R (28) 1000000
+--R Type:
PositiveInteger
+--E 69
+
+--S 70 of 82
m8
+--R
+--R +1000000 2+
+--R (29) | |
+--R + 3 4+
+--R Type: Matrix
Integer
+--E 70
+
+--S 71 of 82
m9
+--R
+--R +1000000 2+
+--R (30) | |
+--R + 3 4+
+--R Type: Matrix
Integer
+--E 71
+
+--S 72 of 82
m10
+--R
+--R +1 2+
+--R (31) | |
+--R +3 4+
+--R Type: Matrix
Integer
+--E 72
-- Input for page EigenPage
)clear all
+--S 73 of 82
m1 : MATRIX FRAC INT := [[1,2,1],[2,1,-2],[1,-2,4]]
+--R
+--R +1 2 1 +
+--R | |
+--R (1) |2 1 - 2|
+--R | |
+--R +1 - 2 4 +
+--R Type: Matrix Fraction
Integer
+--E 73
+
+--S 74 of 82
leig := eigenvalues(m1)
+--R
+--R 2
+--I (2) [5,%A | %A - %A - 5]
+--RType: List Union(Fraction Polynomial Integer,SuchThat(Symbol,Polynomial
Integer))
+--E 74
+
+--S 75 of 82
eigenvector(first(leig),m1)
+--R
+--R + 0 +
+--R | |
+--R | 1|
+--R (3) [|- -|]
+--R | 2|
+--R | |
+--R + 1 +
+--R Type: List Matrix Fraction Polynomial Fraction
Integer
+--E 75
+
+--S 76 of 82
eigenvectors(m1)
+--R
+--R (4)
+--R + 0 +
+--R | |
+--R | 1|
+--R [[eigval= 5,eigmult= 1,eigvec= [|- -|]],
+--R | 2|
+--R | |
+--R + 1 +
+--R +%C+
+--R 2 | |
+--R [eigval= (%C | %C - %C - 5),eigmult= 1,eigvec= [|2 |]]]
+--R | |
+--R +1 +
+--RType: List Record(eigval: Union(Fraction Polynomial
Integer,SuchThat(Symbol,Polynomial Integer)),eigmult:
NonNegativeInteger,eigvec: List Matrix Fraction Polynomial Integer)
+--E 76
+
+--S 77 of 82
radicalEigenvectors(m1)
+--R
+--R (5)
+--R + +--+ +
+--R +--+ |\|21 + 1|
+--R \|21 + 1 |---------|
+--R [[radval= ---------,radmult= 1,radvect= [| 2 |]],
+--R 2 | |
+--R | 2 |
+--R | |
+--R + 1 +
+--R + +--+ +
+--R +--+ |- \|21 + 1|
+--R - \|21 + 1 |-----------|
+--R [radval= -----------,radmult= 1,radvect= [| 2 |]],
+--R 2 | |
+--R | 2 |
+--R | |
+--R + 1 +
+--R + 0 +
+--R | |
+--R | 1|
+--R [radval= 5,radmult= 1,radvect= [|- -|]]]
+--R | 2|
+--R | |
+--R + 1 +
+--RType: List Record(radval: Expression Integer,radmult: Integer,radvect: List
Matrix Expression Integer)
+--E 77
+
+--S 78 of 82
eigenMatrix(m1)
+--R
+--R + +--+ +--+ +
+--R |\|21 + 1 - \|21 + 1 |
+--R |--------- ----------- 0 |
+--R | 2 2 |
+--R (6) | |
+--R | 1|
+--R | 2 2 - -|
+--R | 2|
+--R | |
+--R + 1 1 1 +
+--R Type: Union(Matrix Expression
Integer,...)
+--E 78
+
+--S 79 of 82
m2 : MATRIX FRAC INT := [[-5,-2],[18,7]]
+--R
+--R +- 5 - 2+
+--R (7) | |
+--R +18 7 +
+--R Type: Matrix Fraction
Integer
+--E 79
+
+--S 80 of 82
eigenMatrix(m2)
+--R
+--R (8) "failed"
+--R Type:
Union("failed",...)
+--E 80
+
+--S 81 of 82
m3 : MATRIX FRAC INT := [[1,2],[2,1]]
+--R
+--R +1 2+
+--R (9) | |
+--R +2 1+
+--R Type: Matrix Fraction
Integer
+--E 81
+
+--S 82 of 82
orthonormalBasis(m3)
+--R
+--R + 1 + + 1 +
+--R |- ----| |----|
+--R | +-+| | +-+|
+--R | \|2 | |\|2 |
+--R (10) [| |,| |]
+--R | 1 | | 1 |
+--R | ---- | |----|
+--R | +-+ | | +-+|
+--R + \|2 + +\|2 +
+--R Type: List Matrix Expression
Integer
+--E 82
+
+)spool
+)lisp (bye)
+
@
\eject
\begin{thebibliography}{99}
diff --git a/src/input/overload.input.pamphlet
b/src/input/overload.input.pamphlet
new file mode 100644
index 0000000..2db85a7
--- /dev/null
+++ b/src/input/overload.input.pamphlet
@@ -0,0 +1,680 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input overload.input}
+\author{David Cyganski}
+\maketitle
+\begin{abstract}
+Type conversion and overload problems
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+<<*>>=
+)spool overload.output
+)set message test on
+)set message auto off
+)clear all
+@
+\section{Type Conversion and Overload problems}
+Several non-intuitive problems with overloading and type conversions
+arose while developing the biquaternion (see biquat.input) support
+function collection. I have extracted the minimum code set to illustrate
+each of these problems.
+
+\section{Implicit and Explicit Type Conversions}
+We begin by illustrating function calling with variously typed
+arguments and conversion which we will break in various ways, some
+understandable, some not(?), below.
+
+The cos function will produce float outcomes for float arguments
+<<*>>=
+
+--S 1 of 28
+cos(1.237)
+--R
+--R
+--R (1) 0.3276321705 9891498386
+--R Type:
Float
+--E 1
+
+@
+can handle expressions that mix floats and integers
+<<*>>=
+
+--S 2 of 28
+cos(1.237/2)
+--R
+--R
+--R (2) 0.8147490934 6341557739
+--R Type:
Float
+--E 2
+
+@
+but will respect an integer expression, as we would want it to,
+by not evaluating
+<<*>>=
+
+--S 3 of 28
+cos(2/3)
+--R
+--R
+--R 2
+--R (3) cos(-)
+--R 3
+--R Type: Expression
Integer
+--E 3
+
+@
+We can coerce the evaluation as a float by forcing the floating point
+evaluation of the division and typing of the outcome in a variety of
+ways. Each of the following forms is effective in some appropriate
+and understandable way. Some act explicitly on the ``/'' operator
+to force a polymorphic choice, others convert the type of the second
+constand in each expression which results in a proper implicit selection
+of which ``/'' definitition to use.
+<<*>>=
+
+--S 4 of 28
+cos(2/3::Float)
+--R
+--R
+--R (4) 0.7858872607 7694800072
+--R Type:
Float
+--E 4
+
+--S 5 of 28
+cos((2/3)::Float)
+--R
+--R
+--R (5) 0.7858872607 7694800072
+--R Type:
Float
+--E 5
+
+--S 6 of 28
+cos(2/3$Float)
+--R
+--R
+--R (6) 0.7858872607 7694800072
+--R Type:
Float
+--E 6
+
+--S 7 of 28
+cos((2/3)$Float)
+--R
+--R
+--R (7) 0.7858872607 7694800072
+--R Type:
Float
+--E 7
+
+--S 8 of 28
+cos(2/address@hidden)
+--R
+--R
+--R (8) 0.7858872607 7694800072
+--R Type:
Float
+--E 8
+
+--S 9 of 28
+cos((2/3)@Float)
+--R
+--R
+--R (9) 0.7858872607 7694800072
+--R Type:
Float
+--E 9
+
+@
+But, as we would expect, it is too late to attempt coercion after the
+fact. Coercion operates ``on the surface and not deeply'' as illustrated
+here.
+<<*>>=
+
+--S 10 of 28
+cos(2/3)::Float
+--R
+--R
+--RDaly Bug
+--R Cannot convert from type Expression Integer to Float for value
+--R 2
+--R cos(-)
+--R 3
+--R
+--E 10
+
+@
+However, there is a real need for a deep coercion operator that operates
+on the innermost atomic constants. Suppose we define
+<<*>>=
+
+--S 11 of 28
+cosf(x:Expression Integer):Expression Integer == 1+cos(x/2)
+--R
+--R Function declaration cosf : Expression Integer -> Expression Integer
+--R has been added to workspace.
+--R Type:
Void
+--E 11
+
+@
+which is an example of a simple function that might be defined in the
+course of typical work. We wish to declare functions as having integer
+based fractions, rather than forming approximate decimal expansions, which
+is preferred for purposes of analytic examination and simplification for
+both the human and the Axiom system. The Axiom book and online resources
+are full of examples in which this choice has been made by the authors
+thanks to the power of this form of expression -- even though it amounts
+to lying to Axiom in many cases as to the ultimate destiny of the function
+being defined. But woe to us if we wish later to evaluate it in a more
+general way because it is a tangled web we weave when we practice to
+decieve.
+<<*>>=
+
+--S 12 of 28
+cosf(2/3)
+--R
+--R Compiling function cosf with type Expression Integer -> Expression
+--R Integer
+--R
+--R 1
+--R (11) cos(-) + 1
+--R 3
+--R Type: Expression
Integer
+--E 12
+
+--S 13 of 28
+cosf((2/3)::Float)
+--R
+--R Conversion failed in the compiled user function cosf .
+--R
+--RDaly Bug
+--R Cannot convert from type Float to Expression Integer for value
+--R 0.6666666666 6666666667
+--R
+--E 13
+
+@
+Thus in effect once we wrap a function around an integer base definition,
+we are stuck and unable to evaluate it as a float later, unlike the core
+basic functions that can be used either way. This forces us to choose the
+Float type throughout at a loss of comprehensibility and analyzability,
+unless we seek to more than double our development type by supplying an
+overloaded integer base and Float base version of {\bf every} step of a
+sequential development of a formula.
+
+Bizarrely, the draw function seems to have the power to override the
+type problem as shown here
+<<*>>=
+
+--S 14 of 28
+--draw(cosf(x),x=0..15)
+--E 14
+
+@
+Why can't we grant this deep coercion power to some new form of floating
+point conversion operation which can be applied at will? If draw has
+this power, why not put it in the hands of the user?
+
+Alternatively, it would be best to have a {\bf mixed} type -- mixed
+= Integer/Float. Like Maple expressions it would leave integers as
+integers and floats as floats, unmolested and treated as generic
+constant quantities with distinguishable parts until an {\bf evalf}
+like function that would force them entirely into the Float type. For
+example, in Maple ``cos(2/3)+1.2323'' remains as is, while in Axiom
+we get
+<<*>>=
+
+--S 15 of 28
+cos(2/3)+1.2323
+--R
+--R
+--R (12) 2.0181872607 769480007
+--R Type: Expression
Float
+--E 15
+
+@
+In a way, Axiom already has a quantity treated like this -- the constant
+%pi is treated as a special float which remains unevaluated and does not
+force combination of itself with an Integer. It simply results in a new
+kind of Integer evaluation of type Pi
+<<*>>=
+
+--S 16 of 28
+3/4+%pi
+--R
+--R
+--R 4%pi + 3
+--R (13) --------
+--R 4
+--R Type:
Pi
+--E 16
+
+@
+\section{Overloading problems}
+Now let's examine properties and problems with overloading.
+
+Define the type $Q$ of Hamiltonian biquaternions
+<<*>>=
+
+--S 17 of 28
+C:=Complex Expression Integer
+--R
+--R
+--R (14) Complex Expression Integer
+--R Type:
Domain
+--E 17
+
+--S 18 of 28
+Q:=Quaternion C
+--R
+--R
+--R (15) Quaternion Complex Expression Integer
+--R Type:
Domain
+--E 18
+
+@
+While developing the support functions, this definition of biquat
+division was introduced to simplify the format of the formulae
+<<*>>=
+
+--S 19 of 28
+((x:Q)/(y:Q)):Q == x*inv(y)
+--R
+--R Function declaration ?/? : (Quaternion Complex Expression Integer,
+--R Quaternion Complex Expression Integer) -> Quaternion Complex
+--R Expression Integer has been added to workspace.
+--R Type:
Void
+--E 19
+
+@
+But is this typed function in any way actually restricted to quaternions?
+On the face, it would appear all is normal, here's an example of
+integer division
+<<*>>=
+
+--S 20 of 28
+x:=15/6
+--R
+--R Compiling function / with type (Quaternion Complex Expression
+--R Integer,Quaternion Complex Expression Integer) -> Quaternion
+--R Complex Expression Integer
+--R
+--R 5
+--R (17) -
+--R 2
+--R Type: Quaternion Complex Expression
Integer
+--E 20
+
+@
+But though the answer was right, the type is now a biquat. If we don't
+notice this, and proceed, some things seem still to act normally,
+for example, no complaint from Axiom with
+<<*>>=
+
+--S 21 of 28
+cos(x)
+--R
+--R
+--R 5
+--R (18) cos(-)
+--R 2
+--R Type: Expression
Integer
+--E 21
+
+@
+Of course, we still get a correct answer with
+<<*>>=
+
+--S 22 of 28
+cos(1.237)
+--R
+--R
+--R (19) 0.3276321705 9891498386
+--R Type:
Float
+--E 22
+
+@
+But let's try to apply this as a simple mixed float/integer function
+<<*>>=
+
+--S 23 of 28
+cos(15.457/6)
+--R
+--R Conversion failed in the compiled user function / .
+--R
+--RDaly Bug
+--R Cannot convert from type Float to Quaternion Complex Expression
+--R Integer for value
+--R 15.457
+--R
+--E 23
+
+@
+Obviously the quaternion version of ``/'' is being invoked despite
+mismatches of the arguments and the supposed overloading in effect.
+Well, what if we built a new cosine function that forced the form of
+the arguments into certain types to avoid mismatch?
+<<*>>=
+
+--S 24 of 28
+c(y:Float):Float == cos(y)
+--R
+--R Function declaration c : Float -> Float has been added to workspace.
+--R Type:
Void
+--E 24
+
+@
+At first this seems to work, we can still evaluate a float
+<<*>>=
+
+--S 25 of 28
+c(1.237)
+--R
+--R Compiling function c with type Float -> Float
+--R
+--R (21) 0.3276321705 9891498386
+--R Type:
Float
+--E 25
+
+@
+and we can even get a float answer when we introduce the integer coercible
+biquat variable value generated from the above.
+<<*>>=
+
+--S 26 of 28
+c(x)
+--R
+--R
+--R (22) - 0.8011436155 4693371483
+--R Type:
Float
+--E 26
+
+@
+But that was only misdirection, because this breaks down for reasonable
+expressions because of the ``/'' operation still not being resolved
+correctly.
+<<*>>=
+
+--S 27 of 28
+c(1.237/2)
+--R
+--R Conversion failed in the compiled user function / .
+--R
+--RDaly Bug
+--R Cannot convert from type Float to Quaternion Complex Expression
+--R Integer for value
+--R 1.237
+--R
+--E 27
+
+@
+Rather than complaining about it, what if we tried the various coercions
+that served to solve the similar type conversion problem we had when
+just dealing with Integer Fraction versus Floats at the top of the page.
+Our results are mixed. Recall that each of the following worked in the
+previous case, producing the correct floating result in each case.
+<<*>>=
+
+--S 28 of 28
+cos(2/3::Float)
+--R
+--R
+--R 2
+--R (23) cos(-)
+--R 3
+--R Type: Expression
Integer
+--E 28
+
+--S 29 of 28
+cos((2/3)::Float)
+--R
+--R
+--R (24) 0.7858872607 7694800072
+--R Type:
Float
+--E 29
+
+--S 30 of 28
+cos(2/3$Float)
+--R
+--R
+--R 2
+--R (25) cos(-)
+--R 3
+--R Type: Expression
Integer
+--E 30
+
+--S 31 of 28
+cos((2/3)$Float)
+--R
+--R
+--R (26) 0.7858872607 7694800072
+--R Type:
Float
+--E 31
+
+--S 32 of 28
+cos(2/address@hidden)
+--R
+--R
+--R 2
+--R (27) cos(-)
+--R 3
+--R Type: Expression
Integer
+--E 32
+
+--S 33 of 28
+cos((2/3)@Float)
+--R
+--R
+--RDaly Bug
+--R An expression involving @ Float actually evaluated to one of type
+--R Quaternion Complex Expression Integer . Perhaps you should use ::
+--R Float .
+--E 33
+
+@
+Try these examples with our type constrained function, which has better
+luck now
+<<*>>=
+
+--S 34 of 28
+c(2/3::Float)
+--R
+--R
+--R (28) 0.7858872607 7694800072
+--R Type:
Float
+--E 34
+
+--S 35 of 28
+c((2/3)::Float)
+--R
+--R
+--R (29) 0.7858872607 7694800072
+--R Type:
Float
+--E 35
+
+--S 36 of 28
+c(2/3$Float)
+--R
+--R
+--R (30) 0.7858872607 7694800072
+--R Type:
Float
+--E 36
+
+--S 37 of 28
+c((2/3)$Float)
+--R
+--R
+--R (31) 0.7858872607 7694800072
+--R Type:
Float
+--E 37
+
+--S 38 of 28
+c(2/address@hidden)
+--R
+--R
+--R (32) 0.7858872607 7694800072
+--R Type:
Float
+--E 38
+
+--S 39 of 28
+c((2/3)@Float)
+--R
+--R
+--RDaly Bug
+--R An expression involving @ Float actually evaluated to one of type
+--R Quaternion Complex Expression Integer . Perhaps you should use ::
+--R Float .
+--E 39
+
+@
+Could the above problems have been avoided by not assigning types
+to the function we defined? Let's repeat the entire example with
+this single change for the function c2
+<<*>>=
+
+--S 40 of 28
+c2(y) == cos(y)
+--R
+--R Type:
Void
+--E 40
+
+--S 41 of 28
+c2(1.237)
+--R
+--R Compiling function c2 with type Float -> Float
+--R
+--R (34) 0.3276321705 9891498386
+--R Type:
Float
+--E 41
+
+--S 42 of 28
+c2(x)
+--R
+--R There are 2 exposed and 6 unexposed library operations named cos
+--R having 1 argument(s) but none was determined to be applicable.
+--R Use HyperDoc Browse, or issue
+--R )display op cos
+--R to learn more about the available operations. Perhaps
+--R package-calling the operation or using coercions on the arguments
+--R will allow you to apply the operation.
+--R Cannot find a definition or applicable library operation named cos
+--R with argument type(s)
+--R Quaternion Complex Expression Integer
+--R
+--R Perhaps you should use "@" to indicate the required return type,
+--R or "$" to specify which version of the function you need.
+--R AXIOM will attempt to step through and interpret the code.
+--R
+--R 5
+--R (35) cos(-)
+--R 2
+--R Type: Expression
Integer
+--E 42
+
+@
+But that was only misdirection, because this breaks down for
+reasonable expressions
+<<*>>=
+
+--S 43 of 28
+c2(1.237/2)
+--R
+--R Conversion failed in the compiled user function / .
+--R
+--RDaly Bug
+--R Cannot convert from type Float to Quaternion Complex Expression
+--R Integer for value
+--R 1.237
+--R
+--E 43
+
+@
+and various attempts at coercion also fail -- compare these results
+to the previous ones
+<<*>>=
+
+--S 44 of 28
+c2(2/3::Float)
+--R
+--R
+--R 2
+--R (36) cos(-)
+--R 3
+--R Type: Expression
Integer
+--E 44
+
+--S 45 of 28
+c2((2/3)::Float)
+--R
+--R
+--R (37) 0.7858872607 7694800072
+--R Type:
Float
+--E 45
+
+--S 46 of 28
+c2(2/3$Float)
+--R
+--R
+--R 2
+--R (38) cos(-)
+--R 3
+--R Type: Expression
Integer
+--E 46
+
+--S 47 of 28
+c2((2/3)$Float)
+--R
+--R
+--R (39) 0.7858872607 7694800072
+--R Type:
Float
+--E 47
+
+--S 48 of 28
+c2(2/address@hidden)
+--R
+--R
+--R 2
+--R (40) cos(-)
+--R 3
+--R Type: Expression
Integer
+--E 48
+
+--S 49 of 28
+c2((2/3)@Float)
+--R
+--R
+--RDaly Bug
+--R An expression involving @ Float actually evaluated to one of type
+--R Quaternion Complex Expression Integer . Perhaps you should use ::
+--R Float .
+--E 49
+
+@
+Lastly, we cannot now use the graph function, draw, on such a function
+since the wrong ``/'' function is used, contrary to the bypassing of
+internal types we saw take place with draw in the example prior to the
+introduction of operator overloading.
+<<*>>=
+
+--S 50 of 28
+--draw(c(x),x=0..15)
+--E 50
+
+@
+{\sl Not safe at any speed}. Most oddly, the ordinary cos() function which
+exposes no ``/'' division Now fails to work with draw despite the fact
+that we just saw it above still working with Integer and Float arguments
+applied directly.
+<<*>>=
+
+--S 51 of 28
+--draw(cos(x),x=0..15)
+--E 51
+
+)spool
+)lisp (bye)
+
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}
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