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last edited 6 years ago by pagani |
Edit detail for SandBoxJacobiDiag revision 1 of 1
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changed: - \begin{spad} )abbrev package JACDIAG JacobiDiagonalisation ++ Author: Kurt Pagani ++ Date Created: Sat Jun 16 23:37:27 CEST 2018 ++ License: BSD ++ References: ++ Jacobi, C.G.J. (1846). "Über ein leichtes Verfahren, die in der Theorie ++ der Säkularstörungen vorkommenden Gleichungen numerisch aufzulösen". ++ Crelle's Journal 30, pages 51–94. ++ https://en.wikipedia.org/wiki/Jacobi_eigenvalue_algorithm ++ Description: ++ The Jacobi eigenvalue algorithm is an iterative method for the calculation ++ of the eigenvalues and eigenvectors of a real symmetric matrix. It is named ++ after Carl Gustav Jacob Jacobi, who first proposed the method in 1846, but ++ only became widely used in the 1950s with the advent of computers. ++ Notes ++ * The tolerance and maximum number of iterations may be changed by the ++ function 'setParams'. Defaults are set in the implementation part. ++ * For the case in which one dimension is an independent subspace, the ++ maximum number of iterations may be reached. If 'checkResult' gives ++ a value smaller than 'eps' then 'jacobi' regularly terminates, otherwise ++ an error message is issued. ++ Usage and Example(s) ++ * See test_jacdiag.input ++ * M:=matrix [[1,1],[1,1]], R:=jacobi(M) ++ R.ev .... eigenvalues ++ R.EV .... eigenvectors as columns of matrix R.EV ++ checkResult(R,M) .... Sum(norm(M*v-l*v)) ++ p:=showParams() ..... [maxit= 10000,tol= 0.1 E -8] ++ p.maxit:=20000 ++ setParams(p) ........ [maxit= 20000,tol= 0.1 E -8] ++ IDEA: [jedit]return eps, iter in RES ? ++ JacobiDiagonalisation() : Exports == Implementation where R ==> Float PI ==> PositiveInteger NN ==> NonNegativeInteger IF ==> R VIF ==> Vector R MIF ==> Matrix R RES ==> Record(ev:VIF,EV:MIF) PAR ==> Record(maxit:PI,tol:R) Exports == with jacobi : MIF -> RES checkResult : (RES,MIF) -> R showParams : () -> PAR setParams : PAR -> PAR Implementation == add eps:R:=0.000000001 maxIter:PI:=10000 showParams():PAR == [maxIter,eps] setParams(p:PAR):PAR == maxIter:=p.maxit eps:=p.tol showParams() maxInd(k:PI,n:NN,S:MIF):PI == m:PI:=k+1 for i in k+2..n repeat if abs(S(k,i)) > abs(S(k,m)) then m:=i::PI return m jacobi(M:MIF):RES == not square? M => error "not square" not symmetric? M => error "not symmetric" S:MIF:=copy M n:NN:=nrows S i:PI; k:PI; l:PI; m:PI state:Integer s:IF;c:IF;t:IF;p:IF;y:IF;d:IF;r:IF ind:List(PI):=[1 for i in 1..n] changed:List Boolean:=[false for i in 1..n] e:VIF:=new(n,0$IF)$VIF E:MIF:=new(n,n,0$IF)$MIF E:=diagonalMatrix [1$IF for i in 1..n] A:IF; B:IF count:Integer:=0 state:=n for k in 1..n repeat ind.k:=maxInd(k::PI,n,S) e.k:=S(k,k) changed.k:=true while (state ~= 0) and (count <= maxIter) repeat m:=1 for k in 2..n-1 repeat if abs(S(k,ind.k)) > abs(S(m,ind.m)) then m:=k::PI -- k:=m l:=ind.m p:=S(k,l) -- y:=(e.l - e.k)/2 d:R:=abs(y)+sqrt(p^2+y^2) r:R:=sqrt(p^2+d^2) c:R:=d/r s:R:=p/r t:R:=p^2/d if y < 0$R then s:=-s t:=-t S(k,l):=0$IF -- y:IF:=e.k e.k:=y-t if changed.k and abs(t)<= eps then changed.k:=false state:=state-1 else if (not changed.k) and abs(t)> eps then changed.k:=true state:=state+1 -- y:IF:=e.l e.l:=y+t if changed.l and abs(t)<= eps then changed.l:=false state:=state-1 else if (not changed.l) and abs(t)> eps then changed.l:=true state:=state+1 -- for i in 1..k-1 repeat A:=S(i,k); B:=S(i,l) S(i,k):=c*A-s*B S(i,l):=s*A+c*B for i in k+1..l-1 repeat A:=S(k,i); B:=S(i,l) S(k,i):=c*A-s*B S(i,l):=s*A+c*B for i in l+1..n repeat A:=S(k,i); B:=S(l,i) S(k,i):=c*A-s*B S(l,i):=s*A+c*B -- for i in 1..n repeat A:=E(i,k); B:=E(i,l) E(i,k):=c*A-s*B E(i,l):=s*A+c*B -- ind.k := maxInd(k,n,S) ind.l := maxInd(l,n,S) -- count:=count+1 --output([count,state]) -- if count > maxIter then if checkResult([e,E]$RES,M) > eps then error("Maximum iteration count reached") return [e,E]$RES checkResult(r:RES,M:MIF):R == n:=nrows M s:R:=0$R for i in 1..n repeat v:VIF:=column(r.EV,i) d:VIF:=M*v - r.ev.i * v s:=s+sqrt(dot(d,d)) return s \end{spad} Test flavours \begin{axiom} -- Unittest .....: Package JacobiDiagonalisation -- Author .......: Kurt Pagani -- Date .........: Sun Jun 17 02:14:10 CEST 2018 -- Expected .....: total tests: 16 -- Reference ....: Gregory R.T, Karney D.L., A collection of matrices for testing computational algorithms, -- ............... Krieger (1978), ISBN 0882756494 )set break resume )expose UnittestCount UnittestAux Unittest --)co jacdiag )expose JACDIAG -- Test Suite testsuite "Package JacobiDiagonalisation" -- Cases testcase "jacobi" -- https://en.wikipedia.org/wiki/Jacobi_eigenvalue_algorithm S:= matrix [[4,-30,60,-35], [-30,300,-675,420], [60,-675,1620,-1050],[-35,420,-1050,700]] M:=map(s+->s::Float, S) R:=jacobi(M) ev1:= vector [0.1666428611_7189046264, 37.1014913651_27658188, 2585.2538109289_223144, 1.4780548447_781369118] EV1:= matrix [[0.7926082911_6404284196, -0.1791862905_3191911245, _ 0.0291933231_7921032382_1, -0.5820756994_9722239026], _ [0.4519231209_0048280218, 0.7419177906_0266858684, _ -0.3287120558_2291241902, 0.3705021850_6710175866], _ [0.3224163985_8196511663, -0.1002281368_8300231266, _ 0.7914111458_4119456574, 0.5095786345_0180583406], _ [0.2521611696_891868645, -0.6382825282_3465850686, _ -0.5145527499_457719896, 0.5140482722_2216914813]] eps:=0.0000000001 floatNull(v:Vector Float):Boolean == sqrt dot(v,v) <= eps vector members(map(abs,R.EV-EV1)) testEquals("floatNull(R.ev-ev1)", "true") testEquals("floatNull(vector members(map(abs,R.EV-EV1)))", "true") -- Gregory R.T, Karney D.L, A collection of matrices for testing -- computational algorithms, Krieger, 1978 -- CHAPTER IV, Example 4.1 S:= matrix [[5,4,1,1], [4,5,1,1], [1,1,4,2],[1,1,2,4]] M:=map(s+->s::Float, S) p:=showParams() p.maxit:=20000 p.tol:=0.00000000001 setParams(p) R1:=jacobi(M) testEquals("checkResult(R1,M)<eps","true") -- Example 4.2 S:= matrix [[6,4,4,1], [4,6,1,4], [4,1,6,4],[1,4,4,6]] M:=map(s+->s::Float, S) R2:=jacobi(M) testEquals("checkResult(R2,M)<eps","true") -- Example 4.3 S:= matrix [[2,1,3,4], [1,-3,1,5], [3,1,6,-2],[4,5,-2,-1]] M:=map(s+->s::Float, S) R3:=jacobi(M) testEquals("checkResult(R3,M)<eps","true") -- Example 4.4 S:= matrix [[5,7,6,5], [7,10,8,7], [6,8,10,9],[5,7,9,10]] M:=map(s+->s::Float, S) R4:=jacobi(M) testEquals("checkResult(R4,M)<eps","true") -- Example 4.5 eps:=0.00000001 -- eps from above too small M:= matrix [[0.81321,-0.00013,0.00014,0.00011,0.00021], _ [-0.00013,0.93125,0.23567,0.41235,0.41632], _ [0.00014,0.23567,0.18765,0.50632,0.30697], _ [0.00011,0.41235,0.50632,0.27605,0.46322], _ [0.00021,0.41632,0.30697,0.46322,0.41931]] R5:=jacobi(M) testEquals("checkResult(R5,M)<eps","true") -- Example 4.6 eps:=0.0000001 -- eps from above too small S:= matrix [[5,4,3,2,1],[4,6,0,4,3],[3,0,7,6,5],[2,4,6,8,7],[1,3,5,7,9]] M:=map(s+->s::Float, S) R6:=jacobi(M) testEquals("checkResult(R6,M)<eps","true") -- Example 4.7 S:= matrix [[10,1,2,3,4],[1,9,-1,2,-3],[2,-1,7,3,-5],[3,2,3,12,-1],_ [4,-3,-5,-1,15]] M:=map(s+->s::Float, S) R7:=jacobi(M) testEquals("checkResult(R7,M)<eps","true") -- Example 4.8 S:= matrix [[5,1,-2,0,-2,5],[1,6,-3,2,0,6],[-2,-3,8,-5,-6,0],[0,2,-5,5,1,-2],_ [-2,0,-6,1,6,-3],[5,6,0,-2,-3,8]] M:=map(s+->s::Float, S) R8:=jacobi(M) testEquals("checkResult(R8,M)<eps","true") -- Example 4.9 S:= matrix [[1,2,3,0,1,2],[2,4,5,-1,0,3],[3,5,6,-2,-3,0],[0,-1,-2,1,2,3],_ [1,0,-3,2,4,5],[2,3,0,3,5,6]] M:=map(s+->s::Float, S) R9:=jacobi(M) testEquals("checkResult(R9,M)<eps","true") -- Example 4.10 eps:=0.00001 -- eps from above too small S:= matrix [[611,196,-192,407,-8,-52,-49,29], _ [196,899,113,-192,-71,-43,-8,-44], _ [-192,113,899,196,61,49,8,52], _ [407,-192,196,611,8,44,59,-23],_ [-8,-71,61,8,411,-599,208,208], _ [-52,-43,49,44,-599,411,208,208], _ [-49,-8,8,59,208,208,99,-911], _ [29,-44,52,-23,208,208,-911,99]] M:=map(s+->s::Float, S) R10:=jacobi(M) testEquals("checkResult(R10,M)<eps","true") -- Example 4.11 eps:=0.00001 -- eps from above too small S:= diagonalMatrix [5,6,6,6,6,6,6,6,6,6,5] for i in 2..11 repeat S(i-1,i):=S(i,i-1):=3 for i in 3..11 repeat S(i-2,i):=S(i,i-2):=1 for i in 4..11 repeat S(i-3,i):=S(i,i-3):=1 S(1,2):=S(2,1):=S(10,11):=S(11,10):=2 M:=map(s+->s::Float, S) R11:=jacobi(M) testEquals("checkResult(R11,M)<eps","true") -- Example 4.12 M:=matrix [[0.2500,0.06675,0.04000,0.02475,0.07050,0.06375,0.06925,0.02050,_ 0.03600, -0.01025,-0.00175,0.02750,0.02300,0.00200], _ [0.06675,0.25000,0.10400,0.07475,0.03625,0.11675,0.11050,0.06225,0.05100, _ 0.03250,0.02400,0.03600,0.06350,0.05300], _ [0.04000,0.10400,0.25000,0.14575,0.03725,0.07175,0.07800,0.12200,0.11275, _ 0.09375,0.10175,0.09600,0.14300,0.11550], _ [0.02475,0.07475,0.14575,0.25000,0.05375,0.07000,0.05225,0.12800,0.12475, _ 0.10550,0.13000,0.14575,0.13975,0.13375], _ [0.07050,0.03625,0.03725,0.05375,0.25000,0.04575,0.05750,0.05700,0.05050, _ 0.01475,0.04500,0.07150,0.05300,0.01600], _ [0.06375,0.11675,0.07175,0.07000,0.04575,0.25000,0.08625,0.08800,0.07150, _ 0.04850,0.03200,0.04475,0.03300,0.04500], _ [0.06925,0.11050,0.07800,0.05225,0.05750,0.08625,0.25000,0.08725,0.06950, _ 0.03725,0.04025,0.04300,0.04075,0.01450], _ [0.02050,0.06225,0.12200,0.12800,0.05700,0.08800,0.08725,0.25000,0.14100, _ 0.13275,0.15550,0.13050,0.11825,0.09125], _ [0.03600,0.05100,0.11275,0.12475,0.05050,0.07150,0.06950,0.14100,0.25000, _ 0.07425,0.10750,0.09175,0.10725,0.08225], _ [-0.01025,0.03250,0.09375,0.10550,0.01475,0.04850,0.03725,0.13275,0.07425, _ 0.25000,0.15500,0.09625,0.09950,0.09425], _ [-0.00175,0.02400,0.10175,0.13000,0.04500,0.03200,0.04025,0.15550,0.10750, _ 0.15500,0.25000,0.13350,0.14850,0.13050], _ [0.02750,0.03600,0.09600,0.14575,0.07150,0.04475,0.04300,0.13050,0.09175, _ 0.09625,0.13350,0.25000,0.11100,0.10075], _ [0.02300,0.06350,0.14300,0.13975,0.05300,0.03300,0.04075,0.11825,0.10725, _ 0.09950,0.14850,0.11100,0.25000,0.14325], _ [0.00200,0.05300,0.11550,0.13375,0.01600,0.04500,0.01450,0.09125,0.08225, _ 0.09425,0.13050,0.10075,0.14325,0.25000]] R12:=jacobi(M) testEquals("checkResult(R12,M)<eps","true") -- Example 4.13 (Hilbert Matrix) H(n) == matrix [[1/(i+j-1) for i in 1..n] for j in 1..n] M:=H(20); R13:=jacobi(M) testEquals("checkResult(R13,M)<eps","true") -- Example 4.23 M:=matrix [[1 for i in 1..50] for j in 1..50]; R23:=jacobi(M); testEquals("checkResult(R23,M)<eps","true") -- Results/Statistics )set output algebra on )version )lisp (lisp-implementation-type) )lisp (lisp-implementation-version) statistics() \end{axiom}
fricas
(1) -> <spad>
fricas
)abbrev package JACDIAG JacobiDiagonalisation ++ Author: Kurt Pagani ++ Date Created: Sat Jun 16 23:37:27 CEST 2018 ++ License: BSD ++ References: ++ Jacobi,C.G.J. (1846). "Über ein leichtes Verfahren, die in der Theorie ++ der Säkularstörungen vorkommenden Gleichungen numerisch aufzulösen". ++ Crelle's Journal 30, pages 51–94. ++ https://en.wikipedia.org/wiki/Jacobi_eigenvalue_algorithm ++ Description: ++ The Jacobi eigenvalue algorithm is an iterative method for the calculation ++ of the eigenvalues and eigenvectors of a real symmetric matrix. It is named ++ after Carl Gustav Jacob Jacobi, who first proposed the method in 1846, but ++ only became widely used in the 1950s with the advent of computers. ++ Notes ++ * The tolerance and maximum number of iterations may be changed by the ++ function 'setParams'. Defaults are set in the implementation part. ++ * For the case in which one dimension is an independent subspace, the ++ maximum number of iterations may be reached. If 'checkResult' gives ++ a value smaller than 'eps' then 'jacobi' regularly terminates, otherwise ++ an error message is issued. ++ Usage and Example(s) ++ * See test_jacdiag.input ++ * M:=matrix [[1, 1], [1, 1]], R:=jacobi(M) ++ R.ev .... eigenvalues ++ R.EV .... eigenvectors as columns of matrix R.EV ++ checkResult(R, M) .... Sum(norm(M*v-l*v)) ++ p:=showParams() ..... [maxit= 10000, tol= 0.1 E -8] ++ p.maxit:=20000 ++ setParams(p) ........ [maxit= 20000, tol= 0.1 E -8] ++ IDEA: [jedit]return eps, iter in RES ? ++ JacobiDiagonalisation() : Exports == Implementation where
R ==> Float PI ==> PositiveInteger NN ==> NonNegativeInteger IF ==> R VIF ==> Vector R MIF ==> Matrix R RES ==> Record(ev:VIF,EV:MIF) PAR ==> Record(maxit:PI, tol:R)
Exports == with
jacobi : MIF -> RES checkResult : (RES,MIF) -> R showParams : () -> PAR setParams : PAR -> PAR
Implementation == add
eps:R:=0.000000001 maxIter:PI:=10000
showParams():PAR == [maxIter,eps] setParams(p:PAR):PAR == maxIter:=p.maxit eps:=p.tol showParams()
maxInd(k:PI,n:NN, S:MIF):PI == m:PI:=k+1 for i in k+2..n repeat if abs(S(k, i)) > abs(S(k, m)) then m:=i::PI return m
jacobi(M:MIF):RES == not square? M => error "not square" not symmetric? M => error "not symmetric" S:MIF:=copy M n:NN:=nrows S i:PI; k:PI; l:PI; m:PI state:Integer s:IF;c:IF;t:IF;p:IF;y:IF;d:IF;r:IF ind:List(PI):=[1 for i in 1..n] changed:List Boolean:=[false for i in 1..n] e:VIF:=new(n,0$IF)$VIF E:MIF:=new(n, n, 0$IF)$MIF E:=diagonalMatrix [1$IF for i in 1..n] A:IF; B:IF count:Integer:=0 state:=n for k in 1..n repeat ind.k:=maxInd(k::PI, n, S) e.k:=S(k, k) changed.k:=true while (state ~= 0) and (count <= maxIter) repeat m:=1 for k in 2..n-1 repeat if abs(S(k, ind.k)) > abs(S(m, ind.m)) then m:=k::PI -- k:=m l:=ind.m p:=S(k, l) -- y:=(e.l - e.k)/2 d:R:=abs(y)+sqrt(p^2+y^2) r:R:=sqrt(p^2+d^2) c:R:=d/r s:R:=p/r t:R:=p^2/d if y < 0$R then s:=-s t:=-t S(k, l):=0$IF -- y:IF:=e.k e.k:=y-t if changed.k and abs(t)<= eps then changed.k:=false state:=state-1 else if (not changed.k) and abs(t)> eps then changed.k:=true state:=state+1 -- y:IF:=e.l e.l:=y+t if changed.l and abs(t)<= eps then changed.l:=false state:=state-1 else if (not changed.l) and abs(t)> eps then changed.l:=true state:=state+1 -- for i in 1..k-1 repeat A:=S(i, k); B:=S(i, l) S(i, k):=c*A-s*B S(i, l):=s*A+c*B for i in k+1..l-1 repeat A:=S(k, i); B:=S(i, l) S(k, i):=c*A-s*B S(i, l):=s*A+c*B for i in l+1..n repeat A:=S(k, i); B:=S(l, i) S(k, i):=c*A-s*B S(l, i):=s*A+c*B -- for i in 1..n repeat A:=E(i, k); B:=E(i, l) E(i, k):=c*A-s*B E(i, l):=s*A+c*B -- ind.k := maxInd(k, n, S) ind.l := maxInd(l, n, S) -- count:=count+1 --output([count, state]) -- if count > maxIter then if checkResult([e, E]$RES, M) > eps then error("Maximum iteration count reached") return [e, E]$RES
checkResult(r:RES,M:MIF):R == n:=nrows M s:R:=0$R for i in 1..n repeat v:VIF:=column(r.EV, i) d:VIF:=M*v - r.ev.i * v s:=s+sqrt(dot(d, d)) return s</spad>
fricas
Compiling FriCAS source code from file
/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6878434841567212350-25px001.spad
using old system compiler.
JACDIAG abbreviates package JacobiDiagonalisation
------------------------------------------------------------------------
initializing NRLIB JACDIAG for JacobiDiagonalisation
compiling into NRLIB JACDIAG
compiling exported showParams : () -> Record(maxit: PositiveInteger,
compiling exported setParams : Record(maxit: PositiveInteger,
compiling local maxInd : (PositiveInteger,
compiling exported jacobi : Matrix Float -> Record(ev: Vector Float,
compiling exported checkResult : (Record(ev: Vector Float,
(time taken in buildFunctor: 0)
;;; *** |JacobiDiagonalisation| REDEFINED
;;; *** |JacobiDiagonalisation| REDEFINED
Time: 0 SEC.
Warnings:
[1] setParams: maxit has no value
[2] setParams: tol has no value
[3] maxInd: m has no value
[4] jacobi: m has no value
[5] jacobi: t has no value
[6] jacobi: state has no value
[7] jacobi: s has no value
[8] checkResult: EV has no value
[9] checkResult: ev has no value
Cumulative Statistics for Constructor JacobiDiagonalisation
Time: 0.07 seconds
finalizing NRLIB JACDIAG
Processing JacobiDiagonalisation for Browser database:
--------constructor---------
--->-->JacobiDiagonalisation((jacobi ((Record (: ev (Vector (Float))) (: EV (Matrix (Float)))) (Matrix (Float))))): Not documented!!!!
--->-->JacobiDiagonalisation((checkResult ((Float) (Record (: ev (Vector (Float))) (: EV (Matrix (Float)))) (Matrix (Float))))): Not documented!!!!
--->-->JacobiDiagonalisation((showParams ((Record (: maxit (PositiveInteger)) (: tol (Float)))))): Not documented!!!!
--->-->JacobiDiagonalisation((setParams ((Record (: maxit (PositiveInteger)) (: tol (Float))) (Record (: maxit (PositiveInteger)) (: tol (Float)))))): Not documented!!!!
; compiling file "/var/aw/var/LatexWiki/JACDIAG.NRLIB/JACDIAG.lsp" (written 25 NOV 2024 06:22:00 PM):
; wrote /var/aw/var/LatexWiki/JACDIAG.NRLIB/JACDIAG.fasl
; compilation finished in 0:00:00.704
------------------------------------------------------------------------
JacobiDiagonalisation is now explicitly exposed in frame initial
JacobiDiagonalisation will be automatically loaded when needed from
/var/aw/var/LatexWiki/JACDIAG.NRLIB/JACDIAGTest flavours
- fricas
-- Unittest .....: Package JacobiDiagonalisation
- Author .......: Kurt Pagani
-- Date .........: Sun Jun 17 02:14:10 CEST 2018
-- Expected .....: total tests: 16
-- Reference ....: Gregory R.T,
Karney D.L., A collection of matrices for testing computational algorithms, -- ............... Krieger (1978), ISBN 0882756494 fricas)set break resume
fricas)expose UnittestCount UnittestAux Unittest
UnittestCount is now explicitly exposed in frame initial UnittestAux is now explicitly exposed in frame initial Unittest is now explicitly exposed in frame initial
--)co jacdiagfricas)expose JACDIAG
JacobiDiagonalisation is already explicitly exposed in frame initial
-- Test Suite testsuite "Package JacobiDiagonalisation"
All user variables and function definitions have been cleared. WARNING: string for testsuite should have less than 15 characters!Type: Voidfricas-- Cases testcase "jacobi"
All user variables and function definitions have been cleared.Type: Voidfricas-- https://en.wikipedia.org/wiki/Jacobi_eigenvalue_algorithm S:= matrix [[4,
-30, 60, -35], [-30, 300, -675, 420], [60, -675, 1620, -1050], [-35, 420, -1050, 700]] ![\label{eq1}\left[
\begin{array}{cccc}
4 & -{30}&{60}& -{35}
\
-{30}&{300}& -{675}&{420}
\
{60}& -{675}&{1620}& -{1050}
\
-{35}&{420}& -{1050}&{700}](images/4322683195129483139-16.0px.png "\label{eq1}\left[
\begin{array}{cccc}
4 & -{30}&{60}& -{35}
\
-{30}&{300}& -{675}&{420}
\
{60}& -{675}&{1620}& -{1050}
\
-{35}&{420}& -{1050}&{700}")
(1) Type: Matrix(Integer)fricasM:=map(s+->s::Float,
S) ![\label{eq2}\left[
\begin{array}{cccc}
{4.0}& -{30.0}&{60.0}& -{35.0}
\
-{30.0}&{300.0}& -{675.0}&{420.0}
\
{60.0}& -{675.0}&{1620.0}& -{1050.0}
\
-{35.0}&{420.0}& -{1050.0}&{700.0}](images/2827645992318415605-16.0px.png "\label{eq2}\left[
\begin{array}{cccc}
{4.0}& -{30.0}&{60.0}& -{35.0}
\
-{30.0}&{300.0}& -{675.0}&{420.0}
\
{60.0}& -{675.0}&{1620.0}& -{1050.0}
\
-{35.0}&{420.0}& -{1050.0}&{700.0}")
(2) Type: Matrix(Float)fricasR:=jacobi(M)
![\label{eq3}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[{0.1666428611 \<u> 7189046264}, \:{37.1014913651 \</u> 27658
188}, \: \right.
\
\
\displaystyle
\left.{2585.2538109289 \<u> 223144}, \:{1.4780548447 \</u> 7813691
18}\right]](images/4626403636978500443-16.0px.png "\label{eq3}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[{0.1666428611 \<u> 7189046264}, \:{37.1014913651 \</u> 27658
188}, \: \right.
\
\
\displaystyle
\left.{2585.2538109289 \<u> 223144}, \:{1.4780548447 \</u> 7813691
18}\right]")
(3) Type: Record(ev: Vector(Float),EV: Matrix(Float)) fricasev1:= vector [0.1666428611_7189046264,
37.1014913651_27658188, 2585.2538109289_223144, 1.4780548447_781369118] ![\label{eq4}\begin{array}{@{}l}
\displaystyle
\left[{0.1666428611 \<u> 7189046264}, \:{37.1014913651 \</u> 27658
188}, \: \right.
\
\
\displaystyle
\left.{2585.2538109289 \<u> 223144}, \:{1.4780548447 \</u> 7813691
18}\right]](images/2284469590778659438-16.0px.png "\label{eq4}\begin{array}{@{}l}
\displaystyle
\left[{0.1666428611 \<u> 7189046264}, \:{37.1014913651 \</u> 27658
188}, \: \right.
\
\
\displaystyle
\left.{2585.2538109289 \<u> 223144}, \:{1.4780548447 \</u> 7813691
18}\right]")
(4) Type: Vector(Float)fricasEV1:= matrix [[0.7926082911_6404284196,
-0.1791862905_3191911245, _ 0.0291933231_7921032382_1, -0.5820756994_9722239026], _ [0.4519231209_0048280218, 0.7419177906_0266858684, _ -0.3287120558_2291241902, 0.3705021850_6710175866], _ [0.3224163985_8196511663, -0.1002281368_8300231266, _ 0.7914111458_4119456574, 0.5095786345_0180583406], _ [0.2521611696_891868645, -0.6382825282_3465850686, _ -0.5145527499_457719896, 0.5140482722_2216914813]] ![\label{eq5}\left[
\begin{array}{cccc}
{0.7926082911 \<u> 6404284196}& -{0.1791862905 \</u> 3191911245}&{0.0
291933231 \<u> 7921032382 \</u> 1}& -{0.5820756994 \<u> 9722239026}
\
{0.4519231209 \</u> 0048280218}&{0.7419177906 \<u> 0266858684}& -{0.3
287120558 \</u> 2291241902}&{0.3705021850 \<u> 6710175866}
\
{0.3224163985 \</u> 8196511663}& -{0.1002281368 \<u> 8300231266}&{0.7
914111458 \</u> 4119456574}&{0.5095786345 \<u> 0180583406}
\
{0.2521611696 \</u> 891868645}& -{0.6382825282 \<u> 3465850686}& -{0.5145527499 \</u> 457719896}&{0.5140482722 \_ 2216914813}](images/4636626315701319109-16.0px.png "\label{eq5}\left[
\begin{array}{cccc}
{0.7926082911 \<u> 6404284196}& -{0.1791862905 \</u> 3191911245}&{0.0
291933231 \<u> 7921032382 \</u> 1}& -{0.5820756994 \<u> 9722239026}
\
{0.4519231209 \</u> 0048280218}&{0.7419177906 \<u> 0266858684}& -{0.3
287120558 \</u> 2291241902}&{0.3705021850 \<u> 6710175866}
\
{0.3224163985 \</u> 8196511663}& -{0.1002281368 \<u> 8300231266}&{0.7
914111458 \</u> 4119456574}&{0.5095786345 \<u> 0180583406}
\
{0.2521611696 \</u> 891868645}& -{0.6382825282 \<u> 3465850686}& -{0.5145527499 \</u> 457719896}&{0.5140482722 \_ 2216914813}")
(5) Type: Matrix(Float)fricaseps:=0.0000000001
(6) Type: FloatfricasfloatNull(v:Vector Float):Boolean == sqrt dot(v,
v) <= eps
Function declaration floatNull : Vector(Float) -> Boolean has been added to workspace.Type: Voidfricasvector members(map(abs,
R.EV-EV1)) ![\label{eq7}\begin{array}{@{}l}
\displaystyle
\left[{0.0}, \:{0.0}, \:{0.3 E - 21}, \:{0.0}, \:{0.0}, \:{0.3 E - 20}, \:{0.3 E - 20}, \:{0.5 E - 20}, \right.
\
\
\displaystyle
\left.\:{0.2 E - 20}, \:{0.3 E - 20}, \:{0.0}, \:{0.0}, \:{0.3 E - 20}, \:{0.3 E - 20}, \:{0.0}, \: \right.
\
\
\displaystyle
\left.{0.3 E - 20}\right]](images/7159717639118814391-16.0px.png "\label{eq7}\begin{array}{@{}l}
\displaystyle
\left[{0.0}, \:{0.0}, \:{0.3 E - 21}, \:{0.0}, \:{0.0}, \:{0.3 E - 20}, \:{0.3 E - 20}, \:{0.5 E - 20}, \right.
\
\
\displaystyle
\left.\:{0.2 E - 20}, \:{0.3 E - 20}, \:{0.0}, \:{0.0}, \:{0.3 E - 20}, \:{0.3 E - 20}, \:{0.0}, \: \right.
\
\
\displaystyle
\left.{0.3 E - 20}\right]")
(7) Type: Vector(Float)fricastestEquals("floatNull(R.ev-ev1)","true") fricasCompiling function floatNull with type Vector(Float) -> Boolean
Type: VoidfricastestEquals("floatNull(vector members(map(abs,R.EV-EV1)))", "true") Type: Voidfricas-- Gregory R.T,
Karney D.L, A collection of matrices for testing -- computational algorithms, Krieger, 1978 -- CHAPTER IV, Example 4.1 S:= matrix [[5, 4, 1, 1], [4, 5, 1, 1], [1, 1, 4, 2], [1, 1, 2, 4]] ![\label{eq8}\left[
\begin{array}{cccc}
5 & 4 & 1 & 1
\
4 & 5 & 1 & 1
\
1 & 1 & 4 & 2
\
1 & 1 & 2 & 4](images/9104290058775531835-16.0px.png "\label{eq8}\left[
\begin{array}{cccc}
5 & 4 & 1 & 1
\
4 & 5 & 1 & 1
\
1 & 1 & 4 & 2
\
1 & 1 & 2 & 4")
(8) Type: Matrix(Integer)fricasM:=map(s+->s::Float,
S) ![\label{eq9}\left[
\begin{array}{cccc}
{5.0}&{4.0}&{1.0}&{1.0}
\
{4.0}&{5.0}&{1.0}&{1.0}
\
{1.0}&{1.0}&{4.0}&{2.0}
\
{1.0}&{1.0}&{2.0}&{4.0}](images/2710135830348878782-16.0px.png "\label{eq9}\left[
\begin{array}{cccc}
{5.0}&{4.0}&{1.0}&{1.0}
\
{4.0}&{5.0}&{1.0}&{1.0}
\
{1.0}&{1.0}&{4.0}&{2.0}
\
{1.0}&{1.0}&{2.0}&{4.0}")
(9) Type: Matrix(Float)fricasp:=showParams()
![\label{eq10}\left[{maxit ={10000}}, \:{tol ={0.1 E - 8}}\right]](images/8377803351130098606-16.0px.png "\label{eq10}\left[{maxit ={10000}}, \:{tol ={0.1 E - 8}}\right]")
(10) Type: Record(maxit: PositiveInteger?,tol: Float) fricasp.maxit:=20000
(11) Type: PositiveInteger?fricasp.tol:=0.00000000001
(12) Type: FloatfricassetParams(p)
![\label{eq13}\left[{maxit ={20000}}, \:{tol ={0.1 E - 10}}\right]](images/2145044270697005402-16.0px.png "\label{eq13}\left[{maxit ={20000}}, \:{tol ={0.1 E - 10}}\right]")
(13) Type: Record(maxit: PositiveInteger?,tol: Float) fricasR1:=jacobi(M)
?}}, \: \right. \ \ \displaystyle \left.{ \begin{array}{@{}l} \displaystyle EV = \ \ \displaystyle {\left[ \begin{array}{cccc} {0.7071067811 \ 865475244}&{0.6324555320 \ 336758664}&{0.0}& -{0.3162277660 \ 168379332} \ -{0.7071067811 \ 865475244}&{0.6324555320 \ 336758664}&{0.0}& -{0.3162277660 \ 168379332} \ {0.0}&{0.3162277660 \ 168379332}&{0.7071067811 \ 865475244}&{0.6 324555320 \ 336758664} \ {0.0}&{0.3162277660 \ 168379332}& -{0.7071067811 \ 86547524 4}&{0.6324555320 \ 336758664} " class="equation" src="images/3192297519243230694-16.0px.png" align="bottom" Style="vertical-align:text-bottom" width="816" height="1056"/>
(14) Type: Record(ev: Vector(Float),EV: Matrix(Float)) fricastestEquals("checkResult(R1,M)<eps", "true") Type: Voidfricas-- Example 4.2 S:= matrix [[6,
4, 4, 1], [4, 6, 1, 4], [4, 1, 6, 4], [1, 4, 4, 6]] ![\label{eq15}\left[
\begin{array}{cccc}
6 & 4 & 4 & 1
\
4 & 6 & 1 & 4
\
4 & 1 & 6 & 4
\
1 & 4 & 4 & 6](images/1415736748405463222-16.0px.png "\label{eq15}\left[
\begin{array}{cccc}
6 & 4 & 4 & 1
\
4 & 6 & 1 & 4
\
4 & 1 & 6 & 4
\
1 & 4 & 4 & 6")
(15) Type: Matrix(Integer)fricasM:=map(s+->s::Float,
S) ![\label{eq16}\left[
\begin{array}{cccc}
{6.0}&{4.0}&{4.0}&{1.0}
\
{4.0}&{6.0}&{1.0}&{4.0}
\
{4.0}&{1.0}&{6.0}&{4.0}
\
{1.0}&{4.0}&{4.0}&{6.0}](images/7485724036953670749-16.0px.png "\label{eq16}\left[
\begin{array}{cccc}
{6.0}&{4.0}&{4.0}&{1.0}
\
{4.0}&{6.0}&{1.0}&{4.0}
\
{4.0}&{1.0}&{6.0}&{4.0}
\
{1.0}&{4.0}&{4.0}&{6.0}")
(16) Type: Matrix(Float)fricasR2:=jacobi(M)
![\label{eq17}\begin{array}{@{}l}
\displaystyle
\left[{ev ={\left[ -{1.0}, \:{5.0}, \:{5.0}, \:{15.0}\right]}}, \: \right.
\
\
\displaystyle
\left.{
\begin{array}{@{}l}
\displaystyle
EV =
\
\
\displaystyle
{\left[
\begin{array}{cccc}
{0.5}&{0.5}&{0.5}&{0.5}
\
-{0.5}&{0.5}& -{0.5}&{0.5}
\
-{0.5}& -{0.5}&{0.5}&{0.5}
\
{0.5}& -{0.5}& -{0.5}&{0.5}](images/6001229711054610559-16.0px.png "\label{eq17}\begin{array}{@{}l}
\displaystyle
\left[{ev ={\left[ -{1.0}, \:{5.0}, \:{5.0}, \:{15.0}\right]}}, \: \right.
\
\
\displaystyle
\left.{
\begin{array}{@{}l}
\displaystyle
EV =
\
\
\displaystyle
{\left[
\begin{array}{cccc}
{0.5}&{0.5}&{0.5}&{0.5}
\
-{0.5}&{0.5}& -{0.5}&{0.5}
\
-{0.5}& -{0.5}&{0.5}&{0.5}
\
{0.5}& -{0.5}& -{0.5}&{0.5}")
(17) Type: Record(ev: Vector(Float),EV: Matrix(Float)) fricastestEquals("checkResult(R2,M)<eps", "true") Type: Voidfricas-- Example 4.3 S:= matrix [[2,
1, 3, 4], [1, -3, 1, 5], [3, 1, 6, -2], [4, 5, -2, -1]] ![\label{eq18}\left[
\begin{array}{cccc}
2 & 1 & 3 & 4
\
1 & - 3 & 1 & 5
\
3 & 1 & 6 & - 2
\
4 & 5 & - 2 & - 1](images/3257029801797382809-16.0px.png "\label{eq18}\left[
\begin{array}{cccc}
2 & 1 & 3 & 4
\
1 & - 3 & 1 & 5
\
3 & 1 & 6 & - 2
\
4 & 5 & - 2 & - 1")
(18) Type: Matrix(Integer)fricasM:=map(s+->s::Float,
S) ![\label{eq19}\left[
\begin{array}{cccc}
{2.0}&{1.0}&{3.0}&{4.0}
\
{1.0}& -{3.0}&{1.0}&{5.0}
\
{3.0}&{1.0}&{6.0}& -{2.0}
\
{4.0}&{5.0}& -{2.0}& -{1.0}](images/283189300104644278-16.0px.png "\label{eq19}\left[
\begin{array}{cccc}
{2.0}&{1.0}&{3.0}&{4.0}
\
{1.0}& -{3.0}&{1.0}&{5.0}
\
{3.0}&{1.0}&{6.0}& -{2.0}
\
{4.0}&{5.0}& -{2.0}& -{1.0}")
(19) Type: Matrix(Float)fricasR3:=jacobi(M)
![\label{eq20}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[ -{1.5731907383 \<u> 035074402}, \: -{8.0285783523 \</u> 965
302993}, \right.
\
\
\displaystyle
\left.\:{7.9329047178 \<u> 700173785}, \:{5.6688643728 \</u> 30020
3611}\right]](images/6138420654790665696-16.0px.png "\label{eq20}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[ -{1.5731907383 \<u> 035074402}, \: -{8.0285783523 \</u> 965
302993}, \right.
\
\
\displaystyle
\left.\:{7.9329047178 \<u> 700173785}, \:{5.6688643728 \</u> 30020
3611}\right]")
(20) Type: Record(ev: Vector(Float),EV: Matrix(Float)) fricastestEquals("checkResult(R3,M)<eps", "true") Type: Voidfricas-- Example 4.4 S:= matrix [[5,
7, 6, 5], [7, 10, 8, 7], [6, 8, 10, 9], [5, 7, 9, 10]] ![\label{eq21}\left[
\begin{array}{cccc}
5 & 7 & 6 & 5
\
7 &{10}& 8 & 7
\
6 & 8 &{10}& 9
\
5 & 7 & 9 &{10}](images/4447600182580258386-16.0px.png "\label{eq21}\left[
\begin{array}{cccc}
5 & 7 & 6 & 5
\
7 &{10}& 8 & 7
\
6 & 8 &{10}& 9
\
5 & 7 & 9 &{10}")
(21) Type: Matrix(Integer)fricasM:=map(s+->s::Float,
S) ![\label{eq22}\left[
\begin{array}{cccc}
{5.0}&{7.0}&{6.0}&{5.0}
\
{7.0}&{10.0}&{8.0}&{7.0}
\
{6.0}&{8.0}&{10.0}&{9.0}
\
{5.0}&{7.0}&{9.0}&{10.0}](images/7080693845172082325-16.0px.png "\label{eq22}\left[
\begin{array}{cccc}
{5.0}&{7.0}&{6.0}&{5.0}
\
{7.0}&{10.0}&{8.0}&{7.0}
\
{6.0}&{8.0}&{10.0}&{9.0}
\
{5.0}&{7.0}&{9.0}&{10.0}")
(22) Type: Matrix(Float)fricasR4:=jacobi(M)
![\label{eq23}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[{0.0101500483 \<u> 9789186807 \</u> 2}, \:{3.8580574559 \_ 4
49508547}, \right.
\
\
\displaystyle
\left.\:{0.8431071498 \<u> 550318408}, \:{30.2886853458 \</u> 0212
5436}\right]](images/3668327170948589632-16.0px.png "\label{eq23}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[{0.0101500483 \<u> 9789186807 \</u> 2}, \:{3.8580574559 \_ 4
49508547}, \right.
\
\
\displaystyle
\left.\:{0.8431071498 \<u> 550318408}, \:{30.2886853458 \</u> 0212
5436}\right]")
(23) Type: Record(ev: Vector(Float),EV: Matrix(Float)) fricastestEquals("checkResult(R4,M)<eps", "true") Type: Voidfricas-- Example 4.5 eps:=0.00000001 -- eps from above too small
(24) Type: FloatfricasM:= matrix [[0.81321,
-0.00013, 0.00014, 0.00011, 0.00021], _ [-0.00013, 0.93125, 0.23567, 0.41235, 0.41632], _ [0.00014, 0.23567, 0.18765, 0.50632, 0.30697], _ [0.00011, 0.41235, 0.50632, 0.27605, 0.46322], _ [0.00021, 0.41632, 0.30697, 0.46322, 0.41931]] ![\label{eq25}\left[
\begin{array}{ccccc}
{0.81321}& -{0.00013}&{0.00014}&{0.00011}&{0.00021}
\
-{0.00013}&{0.93125}&{0.23567}&{0.41235}&{0.41632}
\
{0.00014}&{0.23567}&{0.18765}&{0.50632}&{0.30697}
\
{0.00011}&{0.41235}&{0.50632}&{0.27605}&{0.46322}
\
{0.00021}&{0.41632}&{0.30697}&{0.46322}&{0.41931}](images/4439598629386102171-16.0px.png "\label{eq25}\left[
\begin{array}{ccccc}
{0.81321}& -{0.00013}&{0.00014}&{0.00011}&{0.00021}
\
-{0.00013}&{0.93125}&{0.23567}&{0.41235}&{0.41632}
\
{0.00014}&{0.23567}&{0.18765}&{0.50632}&{0.30697}
\
{0.00011}&{0.41235}&{0.50632}&{0.27605}&{0.46322}
\
{0.00021}&{0.41632}&{0.30697}&{0.46322}&{0.41931}")
(25) Type: Matrix(Float)fricasR5:=jacobi(M)
![\label{eq26}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[{0.8132101714 \<u> 8644666656}, \:{0.4198520858 \</u> 425381
4826}, \: \right.
\
\
\displaystyle
\left.-{0.2990822128 \_ 7531886141}, \: \right.
\
\
\displaystyle
\left.{1.6782798304 \<u> 035586382}, \:{0.0152101251 \</u> 4277540
844 \_ 1}\right]](images/7691480906341204043-16.0px.png "\label{eq26}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[{0.8132101714 \<u> 8644666656}, \:{0.4198520858 \</u> 425381
4826}, \: \right.
\
\
\displaystyle
\left.-{0.2990822128 \_ 7531886141}, \: \right.
\
\
\displaystyle
\left.{1.6782798304 \<u> 035586382}, \:{0.0152101251 \</u> 4277540
844 \_ 1}\right]")
(26) Type: Record(ev: Vector(Float),EV: Matrix(Float)) fricastestEquals("checkResult(R5,M)<eps", "true") Type: Voidfricas-- Example 4.6 eps:=0.0000001 -- eps from above too small
(27) Type: FloatfricasS:= matrix [[5,
4, 3, 2, 1], [4, 6, 0, 4, 3], [3, 0, 7, 6, 5], [2, 4, 6, 8, 7], [1, 3, 5, 7, 9]] ![\label{eq28}\left[
\begin{array}{ccccc}
5 & 4 & 3 & 2 & 1
\
4 & 6 & 0 & 4 & 3
\
3 & 0 & 7 & 6 & 5
\
2 & 4 & 6 & 8 & 7
\
1 & 3 & 5 & 7 & 9](images/5211011367161437871-16.0px.png "\label{eq28}\left[
\begin{array}{ccccc}
5 & 4 & 3 & 2 & 1
\
4 & 6 & 0 & 4 & 3
\
3 & 0 & 7 & 6 & 5
\
2 & 4 & 6 & 8 & 7
\
1 & 3 & 5 & 7 & 9")
(28) Type: Matrix(NonNegativeInteger?)fricasM:=map(s+->s::Float,
S) ![\label{eq29}\left[
\begin{array}{ccccc}
{5.0}&{4.0}&{3.0}&{2.0}&{1.0}
\
{4.0}&{6.0}&{0.0}&{4.0}&{3.0}
\
{3.0}&{0.0}&{7.0}&{6.0}&{5.0}
\
{2.0}&{4.0}&{6.0}&{8.0}&{7.0}
\
{1.0}&{3.0}&{5.0}&{7.0}&{9.0}](images/8802726532319954559-16.0px.png "\label{eq29}\left[
\begin{array}{ccccc}
{5.0}&{4.0}&{3.0}&{2.0}&{1.0}
\
{4.0}&{6.0}&{0.0}&{4.0}&{3.0}
\
{3.0}&{0.0}&{7.0}&{6.0}&{5.0}
\
{2.0}&{4.0}&{6.0}&{8.0}&{7.0}
\
{1.0}&{3.0}&{5.0}&{7.0}&{9.0}")
(29) Type: Matrix(Float)fricasR6:=jacobi(M)
![\label{eq30}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[ -{1.0965951816 \<u> 58696809}, \:{7.5137241542 \</u> 053727
573}, \: \right.
\
\
\displaystyle
\left.{4.8489501203 \<u> 161481508}, \:{1.3270455995 \</u> 5676524
47}, \: \right.
\
\
\displaystyle
\left.{22.4068753075 \_ 80410656}\right]](images/9078831103307405728-16.0px.png "\label{eq30}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[ -{1.0965951816 \<u> 58696809}, \:{7.5137241542 \</u> 053727
573}, \: \right.
\
\
\displaystyle
\left.{4.8489501203 \<u> 161481508}, \:{1.3270455995 \</u> 5676524
47}, \: \right.
\
\
\displaystyle
\left.{22.4068753075 \_ 80410656}\right]")
(30) Type: Record(ev: Vector(Float),EV: Matrix(Float)) fricastestEquals("checkResult(R6,M)<eps", "true") Type: Voidfricas-- Example 4.7 S:= matrix [[10,
1, 2, 3, 4], [1, 9, -1, 2, -3], [2, -1, 7, 3, -5], [3, 2, 3, 12, -1], _ [4, -3, -5, -1, 15]] ![\label{eq31}\left[
\begin{array}{ccccc}
{10}& 1 & 2 & 3 & 4
\
1 & 9 & - 1 & 2 & - 3
\
2 & - 1 & 7 & 3 & - 5
\
3 & 2 & 3 &{12}& - 1
\
4 & - 3 & - 5 & - 1 &{15}](images/7416921082511735396-16.0px.png "\label{eq31}\left[
\begin{array}{ccccc}
{10}& 1 & 2 & 3 & 4
\
1 & 9 & - 1 & 2 & - 3
\
2 & - 1 & 7 & 3 & - 5
\
3 & 2 & 3 &{12}& - 1
\
4 & - 3 & - 5 & - 1 &{15}")
(31) Type: Matrix(Integer)fricasM:=map(s+->s::Float,
S) ![\label{eq32}\left[
\begin{array}{ccccc}
{10.0}&{1.0}&{2.0}&{3.0}&{4.0}
\
{1.0}&{9.0}& -{1.0}&{2.0}& -{3.0}
\
{2.0}& -{1.0}&{7.0}&{3.0}& -{5.0}
\
{3.0}&{2.0}&{3.0}&{12.0}& -{1.0}
\
{4.0}& -{3.0}& -{5.0}& -{1.0}&{15.0}](images/6532519076765916875-16.0px.png "\label{eq32}\left[
\begin{array}{ccccc}
{10.0}&{1.0}&{2.0}&{3.0}&{4.0}
\
{1.0}&{9.0}& -{1.0}&{2.0}& -{3.0}
\
{2.0}& -{1.0}&{7.0}&{3.0}& -{5.0}
\
{3.0}&{2.0}&{3.0}&{12.0}& -{1.0}
\
{4.0}& -{3.0}& -{5.0}& -{1.0}&{15.0}")
(32) Type: Matrix(Float)fricasR7:=jacobi(M)
![\label{eq33}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[{6.9948378304 \<u> 964728165}, \:{9.3655549201 \</u> 0613240
88}, \: \right.
\
\
\displaystyle
\left.{1.6552662077 \<u> 271664822}, \:{15.8089207643 \</u> 904919
67}, \: \right.
\
\
\displaystyle
\left.{19.1754202772 \_ 79736326}\right]](images/3108893098653713553-16.0px.png "\label{eq33}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[{6.9948378304 \<u> 964728165}, \:{9.3655549201 \</u> 0613240
88}, \: \right.
\
\
\displaystyle
\left.{1.6552662077 \<u> 271664822}, \:{15.8089207643 \</u> 904919
67}, \: \right.
\
\
\displaystyle
\left.{19.1754202772 \_ 79736326}\right]")
(33) Type: Record(ev: Vector(Float),EV: Matrix(Float)) fricastestEquals("checkResult(R7,M)<eps", "true") Type: Voidfricas-- Example 4.8 S:= matrix [[5,
1, -2, 0, -2, 5], [1, 6, -3, 2, 0, 6], [-2, -3, 8, -5, -6, 0], [0, 2, -5, 5, 1, -2], _ [-2, 0, -6, 1, 6, -3], [5, 6, 0, -2, -3, 8]] ![\label{eq34}\left[
\begin{array}{cccccc}
5 & 1 & - 2 & 0 & - 2 & 5
\
1 & 6 & - 3 & 2 & 0 & 6
\
- 2 & - 3 & 8 & - 5 & - 6 & 0
\
0 & 2 & - 5 & 5 & 1 & - 2
\
- 2 & 0 & - 6 & 1 & 6 & - 3
\
5 & 6 & 0 & - 2 & - 3 & 8](images/9086423618062220153-16.0px.png "\label{eq34}\left[
\begin{array}{cccccc}
5 & 1 & - 2 & 0 & - 2 & 5
\
1 & 6 & - 3 & 2 & 0 & 6
\
- 2 & - 3 & 8 & - 5 & - 6 & 0
\
0 & 2 & - 5 & 5 & 1 & - 2
\
- 2 & 0 & - 6 & 1 & 6 & - 3
\
5 & 6 & 0 & - 2 & - 3 & 8")
(34) Type: Matrix(Integer)fricasM:=map(s+->s::Float,
S) ![\label{eq35}\left[
\begin{array}{cccccc}
{5.0}&{1.0}& -{2.0}&{0.0}& -{2.0}&{5.0}
\
{1.0}&{6.0}& -{3.0}&{2.0}&{0.0}&{6.0}
\
-{2.0}& -{3.0}&{8.0}& -{5.0}& -{6.0}&{0.0}
\
{0.0}&{2.0}& -{5.0}&{5.0}&{1.0}& -{2.0}
\
-{2.0}&{0.0}& -{6.0}&{1.0}&{6.0}& -{3.0}
\
{5.0}&{6.0}&{0.0}& -{2.0}& -{3.0}&{8.0}](images/5319382348230802906-16.0px.png "\label{eq35}\left[
\begin{array}{cccccc}
{5.0}&{1.0}& -{2.0}&{0.0}& -{2.0}&{5.0}
\
{1.0}&{6.0}& -{3.0}&{2.0}&{0.0}&{6.0}
\
-{2.0}& -{3.0}&{8.0}& -{5.0}& -{6.0}&{0.0}
\
{0.0}&{2.0}& -{5.0}&{5.0}&{1.0}& -{2.0}
\
-{2.0}&{0.0}& -{6.0}&{1.0}&{6.0}& -{3.0}
\
{5.0}&{6.0}&{0.0}& -{2.0}& -{3.0}&{8.0}")
(35) Type: Matrix(Float)fricasR8:=jacobi(M)
![\label{eq36}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[{4.4559896384 \<u> 593662322}, \: -{1.5987342935 \</u> 81359
4058}, \right.
\
\
\displaystyle
\left.\:{16.1427446551 \<u> 21993174}, \:{4.4559896384 \</u> 59366
2321}, \: \right.
\
\
\displaystyle
\left.-{1.5987342935 \<u> 813594058}, \:{16.1427446551 \</u> 21993
174}\right]](images/6404332693698593292-16.0px.png "\label{eq36}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[{4.4559896384 \<u> 593662322}, \: -{1.5987342935 \</u> 81359
4058}, \right.
\
\
\displaystyle
\left.\:{16.1427446551 \<u> 21993174}, \:{4.4559896384 \</u> 59366
2321}, \: \right.
\
\
\displaystyle
\left.-{1.5987342935 \<u> 813594058}, \:{16.1427446551 \</u> 21993
174}\right]")
(36) Type: Record(ev: Vector(Float),EV: Matrix(Float)) fricastestEquals("checkResult(R8,M)<eps", "true") Type: Voidfricas-- Example 4.9 S:= matrix [[1,
2, 3, 0, 1, 2], [2, 4, 5, -1, 0, 3], [3, 5, 6, -2, -3, 0], [0, -1, -2, 1, 2, 3], _ [1, 0, -3, 2, 4, 5], [2, 3, 0, 3, 5, 6]] ![\label{eq37}\left[
\begin{array}{cccccc}
1 & 2 & 3 & 0 & 1 & 2
\
2 & 4 & 5 & - 1 & 0 & 3
\
3 & 5 & 6 & - 2 & - 3 & 0
\
0 & - 1 & - 2 & 1 & 2 & 3
\
1 & 0 & - 3 & 2 & 4 & 5
\
2 & 3 & 0 & 3 & 5 & 6](images/1470849166340770674-16.0px.png "\label{eq37}\left[
\begin{array}{cccccc}
1 & 2 & 3 & 0 & 1 & 2
\
2 & 4 & 5 & - 1 & 0 & 3
\
3 & 5 & 6 & - 2 & - 3 & 0
\
0 & - 1 & - 2 & 1 & 2 & 3
\
1 & 0 & - 3 & 2 & 4 & 5
\
2 & 3 & 0 & 3 & 5 & 6")
(37) Type: Matrix(Integer)fricasM:=map(s+->s::Float,
S) ![\label{eq38}\left[
\begin{array}{cccccc}
{1.0}&{2.0}&{3.0}&{0.0}&{1.0}&{2.0}
\
{2.0}&{4.0}&{5.0}& -{1.0}&{0.0}&{3.0}
\
{3.0}&{5.0}&{6.0}& -{2.0}& -{3.0}&{0.0}
\
{0.0}& -{1.0}& -{2.0}&{1.0}&{2.0}&{3.0}
\
{1.0}&{0.0}& -{3.0}&{2.0}&{4.0}&{5.0}
\
{2.0}&{3.0}&{0.0}&{3.0}&{5.0}&{6.0}](images/9109466647747871049-16.0px.png "\label{eq38}\left[
\begin{array}{cccccc}
{1.0}&{2.0}&{3.0}&{0.0}&{1.0}&{2.0}
\
{2.0}&{4.0}&{5.0}& -{1.0}&{0.0}&{3.0}
\
{3.0}&{5.0}&{6.0}& -{2.0}& -{3.0}&{0.0}
\
{0.0}& -{1.0}& -{2.0}&{1.0}&{2.0}&{3.0}
\
{1.0}&{0.0}& -{3.0}&{2.0}&{4.0}&{5.0}
\
{2.0}&{3.0}&{0.0}&{3.0}&{5.0}&{6.0}")
(38) Type: Matrix(Float)fricasR9:=jacobi(M)
![\label{eq39}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[{0.2849864366 \<u> 7450951293}, \: -{1.6963228483 \</u> 9591
36551}, \right.
\
\
\displaystyle
\left.\:{12.4113364117 \<u> 21404142}, \:{0.2849864366 \</u> 74509
51293}, \: \right.
\
\
\displaystyle
\left.-{1.6963228483 \<u> 959136551}, \:{12.4113364117 \</u> 21404
142}\right]](images/2835557064214644056-16.0px.png "\label{eq39}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[{0.2849864366 \<u> 7450951293}, \: -{1.6963228483 \</u> 9591
36551}, \right.
\
\
\displaystyle
\left.\:{12.4113364117 \<u> 21404142}, \:{0.2849864366 \</u> 74509
51293}, \: \right.
\
\
\displaystyle
\left.-{1.6963228483 \<u> 959136551}, \:{12.4113364117 \</u> 21404
142}\right]")
(39) Type: Record(ev: Vector(Float),EV: Matrix(Float)) fricastestEquals("checkResult(R9,M)<eps", "true") Type: Voidfricas-- Example 4.10 eps:=0.00001 -- eps from above too small
(40) Type: FloatfricasS:= matrix [[611,
196, -192, 407, -8, -52, -49, 29], _ [196, 899, 113, -192, -71, -43, -8, -44], _ [-192, 113, 899, 196, 61, 49, 8, 52], _ [407, -192, 196, 611, 8, 44, 59, -23], _ [-8, -71, 61, 8, 411, -599, 208, 208], _ [-52, -43, 49, 44, -599, 411, 208, 208], _ [-49, -8, 8, 59, 208, 208, 99, -911], _ [29, -44, 52, -23, 208, 208, -911, 99]] ![\label{eq41}\left[
\begin{array}{cccccccc}
{611}&{196}& -{192}&{407}& - 8 & -{52}& -{49}&{29}
\
{196}&{899}&{113}& -{192}& -{71}& -{43}& - 8 & -{44}
\
-{192}&{113}&{899}&{196}&{61}&{49}& 8 &{52}
\
{407}& -{192}&{196}&{611}& 8 &{44}&{59}& -{23}
\
- 8 & -{71}&{61}& 8 &{411}& -{599}&{208}&{208}
\
-{52}& -{43}&{49}&{44}& -{599}&{411}&{208}&{208}
\
-{49}& - 8 & 8 &{59}&{208}&{208}&{99}& -{911}
\
{29}& -{44}&{52}& -{23}&{208}&{208}& -{911}&{99}](images/498069809713004649-16.0px.png "\label{eq41}\left[
\begin{array}{cccccccc}
{611}&{196}& -{192}&{407}& - 8 & -{52}& -{49}&{29}
\
{196}&{899}&{113}& -{192}& -{71}& -{43}& - 8 & -{44}
\
-{192}&{113}&{899}&{196}&{61}&{49}& 8 &{52}
\
{407}& -{192}&{196}&{611}& 8 &{44}&{59}& -{23}
\
- 8 & -{71}&{61}& 8 &{411}& -{599}&{208}&{208}
\
-{52}& -{43}&{49}&{44}& -{599}&{411}&{208}&{208}
\
-{49}& - 8 & 8 &{59}&{208}&{208}&{99}& -{911}
\
{29}& -{44}&{52}& -{23}&{208}&{208}& -{911}&{99}")
(41) Type: Matrix(Integer)fricasM:=map(s+->s::Float,
S) ![\label{eq42}\left[
\begin{array}{cccccccc}
{611.0}&{196.0}& -{192.0}&{407.0}& -{8.0}& -{52.0}& -{49.0}&{2
9.0}
\
{196.0}&{899.0}&{113.0}& -{192.0}& -{71.0}& -{43.0}& -{8.0}& -{44.0}
\
-{192.0}&{113.0}&{899.0}&{196.0}&{61.0}&{49.0}&{8.0}&{52.0}
\
{407.0}& -{192.0}&{196.0}&{611.0}&{8.0}&{44.0}&{59.0}& -{23.0}
\
-{8.0}& -{71.0}&{61.0}&{8.0}&{411.0}& -{599.0}&{208.0}&{208.0}
\
-{52.0}& -{43.0}&{49.0}&{44.0}& -{599.0}&{411.0}&{208.0}&{208.0}
\
-{49.0}& -{8.0}&{8.0}&{59.0}&{208.0}&{208.0}&{99.0}& -{911.0}
\
{29.0}& -{44.0}&{52.0}& -{23.0}&{208.0}&{208.0}& -{911.0}&{99.0}](images/6072480643871296265-16.0px.png "\label{eq42}\left[
\begin{array}{cccccccc}
{611.0}&{196.0}& -{192.0}&{407.0}& -{8.0}& -{52.0}& -{49.0}&{2
9.0}
\
{196.0}&{899.0}&{113.0}& -{192.0}& -{71.0}& -{43.0}& -{8.0}& -{44.0}
\
-{192.0}&{113.0}&{899.0}&{196.0}&{61.0}&{49.0}&{8.0}&{52.0}
\
{407.0}& -{192.0}&{196.0}&{611.0}&{8.0}&{44.0}&{59.0}& -{23.0}
\
-{8.0}& -{71.0}&{61.0}&{8.0}&{411.0}& -{599.0}&{208.0}&{208.0}
\
-{52.0}& -{43.0}&{49.0}&{44.0}& -{599.0}&{411.0}&{208.0}&{208.0}
\
-{49.0}& -{8.0}&{8.0}&{59.0}&{208.0}&{208.0}&{99.0}& -{911.0}
\
{29.0}& -{44.0}&{52.0}& -{23.0}&{208.0}&{208.0}& -{911.0}&{99.0}")
(42) Type: Matrix(Float)fricasR10:=jacobi(M)
![\label{eq43}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[{0.0980486407 \<u> 2151706114 \</u> 7}, \:{999.9999999999 \_ 9999958}, \right.
\
\
\displaystyle
\left.\:{1000.0}, \:{1020.0490184299 \_ 968238}, \: \right.
\
\
\displaystyle
\left.{0.3776805208 \_ 266153749 E - 15}, \: \right.
\
\
\displaystyle
\left.{1019.9999999999 \<u> 989334}, \: -{1020.0490184299 \</u> 96
8238}, \right.
\
\
\displaystyle
\left.\:{1019.9019513592 \_ 795495}\right]](images/1826835855424980529-16.0px.png "\label{eq43}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[{0.0980486407 \<u> 2151706114 \</u> 7}, \:{999.9999999999 \_ 9999958}, \right.
\
\
\displaystyle
\left.\:{1000.0}, \:{1020.0490184299 \_ 968238}, \: \right.
\
\
\displaystyle
\left.{0.3776805208 \_ 266153749 E - 15}, \: \right.
\
\
\displaystyle
\left.{1019.9999999999 \<u> 989334}, \: -{1020.0490184299 \</u> 96
8238}, \right.
\
\
\displaystyle
\left.\:{1019.9019513592 \_ 795495}\right]")
(43) Type: Record(ev: Vector(Float),EV: Matrix(Float)) fricastestEquals("checkResult(R10,M)<eps", "true") Type: Voidfricas-- Example 4.11 eps:=0.00001 -- eps from above too small
(44) Type: FloatfricasS:= diagonalMatrix [5,
6, 6, 6, 6, 6, 6, 6, 6, 6, 5] ![\label{eq45}\left[
\begin{array}{ccccccccccc}
5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\
0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\
0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\
0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\
0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0
\
0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0
\
0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5](images/7115508418202764909-16.0px.png "\label{eq45}\left[
\begin{array}{ccccccccccc}
5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\
0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\
0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\
0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\
0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0
\
0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0
\
0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5")
(45) Type: Matrix(Integer)fricasfor i in 2..11 repeat S(i-1,
i):=S(i, i-1):=3 Type: Voidfricasfor i in 3..11 repeat S(i-2,
i):=S(i, i-2):=1 Type: Voidfricasfor i in 4..11 repeat S(i-3,
i):=S(i, i-3):=1 Type: VoidfricasS(1,
2):=S(2, 1):=S(10, 11):=S(11, 10):=2 
(46) Type: PositiveInteger?fricasM:=map(s+->s::Float,
S) ![\label{eq47}\left[
\begin{array}{ccccccccccc}
{5.0}&{2.0}&{1.0}&{1.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}
\
{2.0}&{6.0}&{3.0}&{1.0}&{1.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}
\
{1.0}&{3.0}&{6.0}&{3.0}&{1.0}&{1.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}
\
{1.0}&{1.0}&{3.0}&{6.0}&{3.0}&{1.0}&{1.0}&{0.0}&{0.0}&{0.0}&{0.0}
\
{0.0}&{1.0}&{1.0}&{3.0}&{6.0}&{3.0}&{1.0}&{1.0}&{0.0}&{0.0}&{0.0}
\
{0.0}&{0.0}&{1.0}&{1.0}&{3.0}&{6.0}&{3.0}&{1.0}&{1.0}&{0.0}&{0.0}
\
{0.0}&{0.0}&{0.0}&{1.0}&{1.0}&{3.0}&{6.0}&{3.0}&{1.0}&{1.0}&{0.0}
\
{0.0}&{0.0}&{0.0}&{0.0}&{1.0}&{1.0}&{3.0}&{6.0}&{3.0}&{1.0}&{1.0}
\
{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{1.0}&{1.0}&{3.0}&{6.0}&{3.0}&{1.0}
\
{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{1.0}&{1.0}&{3.0}&{6.0}&{2.0}
\
{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{1.0}&{1.0}&{2.0}&{5.0}](images/2099065019673493260-16.0px.png "\label{eq47}\left[
\begin{array}{ccccccccccc}
{5.0}&{2.0}&{1.0}&{1.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}
\
{2.0}&{6.0}&{3.0}&{1.0}&{1.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}
\
{1.0}&{3.0}&{6.0}&{3.0}&{1.0}&{1.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}
\
{1.0}&{1.0}&{3.0}&{6.0}&{3.0}&{1.0}&{1.0}&{0.0}&{0.0}&{0.0}&{0.0}
\
{0.0}&{1.0}&{1.0}&{3.0}&{6.0}&{3.0}&{1.0}&{1.0}&{0.0}&{0.0}&{0.0}
\
{0.0}&{0.0}&{1.0}&{1.0}&{3.0}&{6.0}&{3.0}&{1.0}&{1.0}&{0.0}&{0.0}
\
{0.0}&{0.0}&{0.0}&{1.0}&{1.0}&{3.0}&{6.0}&{3.0}&{1.0}&{1.0}&{0.0}
\
{0.0}&{0.0}&{0.0}&{0.0}&{1.0}&{1.0}&{3.0}&{6.0}&{3.0}&{1.0}&{1.0}
\
{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{1.0}&{1.0}&{3.0}&{6.0}&{3.0}&{1.0}
\
{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{1.0}&{1.0}&{3.0}&{6.0}&{2.0}
\
{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{1.0}&{1.0}&{2.0}&{5.0}")
(47) Type: Matrix(Float)fricasR11:=jacobi(M)
![\label{eq48}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[{4.1292484841 \<u> 890931323}, \:{0.5222822874 \</u> 6137264
404}, \: \right.
\
\
\displaystyle
\left.{8.8284271247 \<u> 461905061}, \:{3.9999999999 \</u> 9999999
97}, \: \right.
\
\
\displaystyle
\left.{14.9418193276 \<u> 76381949}, \:{3.1715728752 \</u> 5380770
16}, \: \right.
\
\
\displaystyle
\left.{5.9999999999 \<u> 99999446}, \:{1.8038475772 \</u> 93370916
8}, \: \right.
\
\
\displaystyle
\left.{12.1961524227 \<u> 06631475}, \:{4.4066499006 \</u> 7315222
96}, \: \right.
\
\
\displaystyle
\left.{3.9999999999 \_ 999999999}\right]](images/7836957948002552200-16.0px.png "\label{eq48}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[{4.1292484841 \<u> 890931323}, \:{0.5222822874 \</u> 6137264
404}, \: \right.
\
\
\displaystyle
\left.{8.8284271247 \<u> 461905061}, \:{3.9999999999 \</u> 9999999
97}, \: \right.
\
\
\displaystyle
\left.{14.9418193276 \<u> 76381949}, \:{3.1715728752 \</u> 5380770
16}, \: \right.
\
\
\displaystyle
\left.{5.9999999999 \<u> 99999446}, \:{1.8038475772 \</u> 93370916
8}, \: \right.
\
\
\displaystyle
\left.{12.1961524227 \<u> 06631475}, \:{4.4066499006 \</u> 7315222
96}, \: \right.
\
\
\displaystyle
\left.{3.9999999999 \_ 999999999}\right]")
(48) Type: Record(ev: Vector(Float),EV: Matrix(Float)) fricastestEquals("checkResult(R11,M)<eps", "true") Type: Voidfricas-- Example 4.12
M:=matrix [[0.2500,0.06675, 0.04000, 0.02475, 0.07050, 0.06375, 0.06925, 0.02050, _ 0.03600, -0.01025, -0.00175, 0.02750, 0.02300, 0.00200], _ [0.06675, 0.25000, 0.10400, 0.07475, 0.03625, 0.11675, 0.11050, 0.06225, 0.05100, _ 0.03250, 0.02400, 0.03600, 0.06350, 0.05300], _ [0.04000, 0.10400, 0.25000, 0.14575, 0.03725, 0.07175, 0.07800, 0.12200, 0.11275, _ 0.09375, 0.10175, 0.09600, 0.14300, 0.11550], _ [0.02475, 0.07475, 0.14575, 0.25000, 0.05375, 0.07000, 0.05225, 0.12800, 0.12475, _ 0.10550, 0.13000, 0.14575, 0.13975, 0.13375], _ [0.07050, 0.03625, 0.03725, 0.05375, 0.25000, 0.04575, 0.05750, 0.05700, 0.05050, _ 0.01475, 0.04500, 0.07150, 0.05300, 0.01600], _ [0.06375, 0.11675, 0.07175, 0.07000, 0.04575, 0.25000, 0.08625, 0.08800, 0.07150, _ 0.04850, 0.03200, 0.04475, 0.03300, 0.04500], _ [0.06925, 0.11050, 0.07800, 0.05225, 0.05750, 0.08625, 0.25000, 0.08725, 0.06950, _ 0.03725, 0.04025, 0.04300, 0.04075, 0.01450], _ [0.02050, 0.06225, 0.12200, 0.12800, 0.05700, 0.08800, 0.08725, 0.25000, 0.14100, _ 0.13275, 0.15550, 0.13050, 0.11825, 0.09125], _ [0.03600, 0.05100, 0.11275, 0.12475, 0.05050, 0.07150, 0.06950, 0.14100, 0.25000, _ 0.07425, 0.10750, 0.09175, 0.10725, 0.08225], _ [-0.01025, 0.03250, 0.09375, 0.10550, 0.01475, 0.04850, 0.03725, 0.13275, 0.07425, _ 0.25000, 0.15500, 0.09625, 0.09950, 0.09425], _ [-0.00175, 0.02400, 0.10175, 0.13000, 0.04500, 0.03200, 0.04025, 0.15550, 0.10750, _ 0.15500, 0.25000, 0.13350, 0.14850, 0.13050], _ [0.02750, 0.03600, 0.09600, 0.14575, 0.07150, 0.04475, 0.04300, 0.13050, 0.09175, _ 0.09625, 0.13350, 0.25000, 0.11100, 0.10075], _ [0.02300, 0.06350, 0.14300, 0.13975, 0.05300, 0.03300, 0.04075, 0.11825, 0.10725, _ 0.09950, 0.14850, 0.11100, 0.25000, 0.14325], _ [0.00200, 0.05300, 0.11550, 0.13375, 0.01600, 0.04500, 0.01450, 0.09125, 0.08225, _ 0.09425, 0.13050, 0.10075, 0.14325, 0.25000]] ![\label{eq49}\left[
\begin{array}{cccccccccccccc}
{0.25}&{0.06675}&{0.04}&{0.02475}&{0.0705}&{0.06375}&{0.06925}&{0.0
205}&{0.036}& -{0.01025}& -{0.00175}&{0.0275}&{0.023}&{0.002}
\
{0.06675}&{0.25}&{0.104}&{0.07475}&{0.03625}&{0.11675}&{0.110
5}&{0.06225}&{0.051}&{0.0325}&{0.024}&{0.036}&{0.0635}&{0.053}
\
{0.04}&{0.104}&{0.25}&{0.14575}&{0.03725}&{0.07175}&{0.078}&{0.1
22}&{0.11275}&{0.09375}&{0.10175}&{0.096}&{0.143}&{0.1155}
\
{0.02475}&{0.07475}&{0.14575}&{0.25}&{0.05375}&{0.07}&{0.0522
5}&{0.128}&{0.12475}&{0.1055}&{0.13}&{0.14575}&{0.13975}&{0.1
3375}
\
{0.0705}&{0.03625}&{0.03725}&{0.05375}&{0.25}&{0.04575}&{0.05
75}&{0.057}&{0.0505}&{0.01475}&{0.045}&{0.0715}&{0.053}&{0.01
6}
\
{0.06375}&{0.11675}&{0.07175}&{0.07}&{0.04575}&{0.25}&{0.0862
5}&{0.088}&{0.0715}&{0.0485}&{0.032}&{0.04475}&{0.033}&{0.045}
\
{0.06925}&{0.1105}&{0.078}&{0.05225}&{0.0575}&{0.08625}&{0.25}&{0.0
8725}&{0.0695}&{0.03725}&{0.04025}&{0.043}&{0.04075}&{0.0145}
\
{0.0205}&{0.06225}&{0.122}&{0.128}&{0.057}&{0.088}&{0.08725}&{0.2
5}&{0.141}&{0.13275}&{0.1555}&{0.1305}&{0.11825}&{0.09125}
\
{0.036}&{0.051}&{0.11275}&{0.12475}&{0.0505}&{0.0715}&{0.0695}&{0.1
41}&{0.25}&{0.07425}&{0.1075}&{0.09175}&{0.10725}&{0.08225}
\
-{0.01025}&{0.0325}&{0.09375}&{0.1055}&{0.01475}&{0.0485}&{0.0
3725}&{0.13275}&{0.07425}&{0.25}&{0.155}&{0.09625}&{0.0995}&{0.0
9425}
\
-{0.00175}&{0.024}&{0.10175}&{0.13}&{0.045}&{0.032}&{0.04025}&{0.1
555}&{0.1075}&{0.155}&{0.25}&{0.1335}&{0.1485}&{0.1305}
\
{0.0275}&{0.036}&{0.096}&{0.14575}&{0.0715}&{0.04475}&{0.043}&{0.1
305}&{0.09175}&{0.09625}&{0.1335}&{0.25}&{0.111}&{0.10075}
\
{0.023}&{0.0635}&{0.143}&{0.13975}&{0.053}&{0.033}&{0.04075}&{0.1
1825}&{0.10725}&{0.0995}&{0.1485}&{0.111}&{0.25}&{0.14325}
\
{0.002}&{0.053}&{0.1155}&{0.13375}&{0.016}&{0.045}&{0.0145}&{0.0
9125}&{0.08225}&{0.09425}&{0.1305}&{0.10075}&{0.14325}&{0.25}](images/4186851966546187576-16.0px.png "\label{eq49}\left[
\begin{array}{cccccccccccccc}
{0.25}&{0.06675}&{0.04}&{0.02475}&{0.0705}&{0.06375}&{0.06925}&{0.0
205}&{0.036}& -{0.01025}& -{0.00175}&{0.0275}&{0.023}&{0.002}
\
{0.06675}&{0.25}&{0.104}&{0.07475}&{0.03625}&{0.11675}&{0.110
5}&{0.06225}&{0.051}&{0.0325}&{0.024}&{0.036}&{0.0635}&{0.053}
\
{0.04}&{0.104}&{0.25}&{0.14575}&{0.03725}&{0.07175}&{0.078}&{0.1
22}&{0.11275}&{0.09375}&{0.10175}&{0.096}&{0.143}&{0.1155}
\
{0.02475}&{0.07475}&{0.14575}&{0.25}&{0.05375}&{0.07}&{0.0522
5}&{0.128}&{0.12475}&{0.1055}&{0.13}&{0.14575}&{0.13975}&{0.1
3375}
\
{0.0705}&{0.03625}&{0.03725}&{0.05375}&{0.25}&{0.04575}&{0.05
75}&{0.057}&{0.0505}&{0.01475}&{0.045}&{0.0715}&{0.053}&{0.01
6}
\
{0.06375}&{0.11675}&{0.07175}&{0.07}&{0.04575}&{0.25}&{0.0862
5}&{0.088}&{0.0715}&{0.0485}&{0.032}&{0.04475}&{0.033}&{0.045}
\
{0.06925}&{0.1105}&{0.078}&{0.05225}&{0.0575}&{0.08625}&{0.25}&{0.0
8725}&{0.0695}&{0.03725}&{0.04025}&{0.043}&{0.04075}&{0.0145}
\
{0.0205}&{0.06225}&{0.122}&{0.128}&{0.057}&{0.088}&{0.08725}&{0.2
5}&{0.141}&{0.13275}&{0.1555}&{0.1305}&{0.11825}&{0.09125}
\
{0.036}&{0.051}&{0.11275}&{0.12475}&{0.0505}&{0.0715}&{0.0695}&{0.1
41}&{0.25}&{0.07425}&{0.1075}&{0.09175}&{0.10725}&{0.08225}
\
-{0.01025}&{0.0325}&{0.09375}&{0.1055}&{0.01475}&{0.0485}&{0.0
3725}&{0.13275}&{0.07425}&{0.25}&{0.155}&{0.09625}&{0.0995}&{0.0
9425}
\
-{0.00175}&{0.024}&{0.10175}&{0.13}&{0.045}&{0.032}&{0.04025}&{0.1
555}&{0.1075}&{0.155}&{0.25}&{0.1335}&{0.1485}&{0.1305}
\
{0.0275}&{0.036}&{0.096}&{0.14575}&{0.0715}&{0.04475}&{0.043}&{0.1
305}&{0.09175}&{0.09625}&{0.1335}&{0.25}&{0.111}&{0.10075}
\
{0.023}&{0.0635}&{0.143}&{0.13975}&{0.053}&{0.033}&{0.04075}&{0.1
1825}&{0.10725}&{0.0995}&{0.1485}&{0.111}&{0.25}&{0.14325}
\
{0.002}&{0.053}&{0.1155}&{0.13375}&{0.016}&{0.045}&{0.0145}&{0.0
9125}&{0.08225}&{0.09425}&{0.1305}&{0.10075}&{0.14325}&{0.25}")
(49) Type: Matrix(Float)fricasR12:=jacobi(M)
![\label{eq50}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[{0.1663246021 \<u> 0025823682}, \:{0.1713075617 \</u> 897417
1399}, \: \right.
\
\
\displaystyle
\left.{0.0972092161 \<u> 6203514813 \</u> 8}, \:{0.1227875231 \_ 8
17131926}, \right.
\
\
\displaystyle
\left.\:{0.2677332979 \<u> 5388836645}, \:{0.4627662026 \</u> 9414
64008}, \: \right.
\
\
\displaystyle
\left.{0.1032157624 \_ 3104449922}, \: \right.
\
\
\displaystyle
\left.{0.0643799133 \<u> 6741279490 \</u> 3}, \: \right.
\
\
\displaystyle
\left.{0.1773563338 \_ 2092625442}, \: \right.
\
\
\displaystyle
\left.{0.0735971188 \<u> 5341424867 \</u> 5}, \:{1.3340348369 \_ 5
64426216}, \right.
\
\
\displaystyle
\left.\:{0.1434228761 \_ 4650515836}, \: \right.
\
\
\displaystyle
\left.{0.0842252705 \<u> 6463554604 \</u> 4}, \: \right.
\
\
\displaystyle
\left.{0.2316394839 \_ 7783581793}\right]](images/1386684550116499307-16.0px.png "\label{eq50}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[{0.1663246021 \<u> 0025823682}, \:{0.1713075617 \</u> 897417
1399}, \: \right.
\
\
\displaystyle
\left.{0.0972092161 \<u> 6203514813 \</u> 8}, \:{0.1227875231 \_ 8
17131926}, \right.
\
\
\displaystyle
\left.\:{0.2677332979 \<u> 5388836645}, \:{0.4627662026 \</u> 9414
64008}, \: \right.
\
\
\displaystyle
\left.{0.1032157624 \_ 3104449922}, \: \right.
\
\
\displaystyle
\left.{0.0643799133 \<u> 6741279490 \</u> 3}, \: \right.
\
\
\displaystyle
\left.{0.1773563338 \_ 2092625442}, \: \right.
\
\
\displaystyle
\left.{0.0735971188 \<u> 5341424867 \</u> 5}, \:{1.3340348369 \_ 5
64426216}, \right.
\
\
\displaystyle
\left.\:{0.1434228761 \_ 4650515836}, \: \right.
\
\
\displaystyle
\left.{0.0842252705 \<u> 6463554604 \</u> 4}, \: \right.
\
\
\displaystyle
\left.{0.2316394839 \_ 7783581793}\right]")
(50) Type: Record(ev: Vector(Float),EV: Matrix(Float)) fricastestEquals("checkResult(R12,M)<eps", "true") Type: Voidfricas-- Example 4.13 (Hilbert Matrix)
H(n) == matrix [[1/(i+j-1) for i in 1..n] for j in 1..n]Type: VoidfricasM:=H(20);
fricasCompiling function H with type PositiveInteger -> Matrix(Fraction( Integer))Type: Matrix(Fraction(Integer))fricasR13:=jacobi(M)
![\label{eq51}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[{1.9071347204 \<u> 07253103}, \:{0.4870384065 \</u> 72048867
81}, \: \right.
\
\
\displaystyle
\left.{0.0008676711 \<u> 0917149782 \</u> 705}, \: \right.
\
\
\displaystyle
\left.{0.0000048305 \<u> 1004864734 \</u> 33279}, \: \right.
\
\
\displaystyle
\left.{0.0089611286 \<u> 1485648034 \</u> 98}, \: \right.
\
\
\displaystyle
\left.{0.1590216371 \_ 077614324 E - 11}, \: \right.
\
\
\displaystyle
\left.{0.1413948074 \_ 4628595824 E - 7}, \: \right.
\
\
\displaystyle
\left.{0.3979401165 \_ 5463231616 E - 11}, \: \right.
\
\
\displaystyle
\left.{0.0755958213 \<u> 0544095843 \</u> 9}, \: \right.
\
\
\displaystyle
\left.{0.1668930345 \_ 4225524645 E - 11}, \: \right.
\
\
\displaystyle
\left.{0.1872845593 \_ 6715022839 E - 11}, \: \right.
\
\
\displaystyle
\left.{0.4156604438 \_ 2412458777 E - 11}, \: \right.
\
\
\displaystyle
\left.{0.6029936660 \_ 8059110817 E - 9}, \: \right.
\
\
\displaystyle
\left.{0.1279656833 \_ 3254260344 E - 11}, \: \right.
\
\
\displaystyle
\left.{0.2827652010 \_ 3941708864 E - 6}, \: \right.
\
\
\displaystyle
\left.{0.1328459298 \_ 5173200512 E - 12}, \: \right.
\
\
\displaystyle
\left.{0.9892736125 \_ 7865554553 E - 12}, \: \right.
\
\
\displaystyle
\left.{0.3879726539 \_ 3557307554 E - 11}, \: \right.
\
\
\displaystyle
\left.{0.3758601579 \_ 8175153646 E - 11}, \: \right.
\
\
\displaystyle
\left.{0.0000703343 \<u> 1473192346 \</u> 1446}\right]](images/3615912294114173446-16.0px.png "\label{eq51}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
ev ={
\begin{array}{@{}l}
\displaystyle
\left[{1.9071347204 \<u> 07253103}, \:{0.4870384065 \</u> 72048867
81}, \: \right.
\
\
\displaystyle
\left.{0.0008676711 \<u> 0917149782 \</u> 705}, \: \right.
\
\
\displaystyle
\left.{0.0000048305 \<u> 1004864734 \</u> 33279}, \: \right.
\
\
\displaystyle
\left.{0.0089611286 \<u> 1485648034 \</u> 98}, \: \right.
\
\
\displaystyle
\left.{0.1590216371 \_ 077614324 E - 11}, \: \right.
\
\
\displaystyle
\left.{0.1413948074 \_ 4628595824 E - 7}, \: \right.
\
\
\displaystyle
\left.{0.3979401165 \_ 5463231616 E - 11}, \: \right.
\
\
\displaystyle
\left.{0.0755958213 \<u> 0544095843 \</u> 9}, \: \right.
\
\
\displaystyle
\left.{0.1668930345 \_ 4225524645 E - 11}, \: \right.
\
\
\displaystyle
\left.{0.1872845593 \_ 6715022839 E - 11}, \: \right.
\
\
\displaystyle
\left.{0.4156604438 \_ 2412458777 E - 11}, \: \right.
\
\
\displaystyle
\left.{0.6029936660 \_ 8059110817 E - 9}, \: \right.
\
\
\displaystyle
\left.{0.1279656833 \_ 3254260344 E - 11}, \: \right.
\
\
\displaystyle
\left.{0.2827652010 \_ 3941708864 E - 6}, \: \right.
\
\
\displaystyle
\left.{0.1328459298 \_ 5173200512 E - 12}, \: \right.
\
\
\displaystyle
\left.{0.9892736125 \_ 7865554553 E - 12}, \: \right.
\
\
\displaystyle
\left.{0.3879726539 \_ 3557307554 E - 11}, \: \right.
\
\
\displaystyle
\left.{0.3758601579 \_ 8175153646 E - 11}, \: \right.
\
\
\displaystyle
\left.{0.0000703343 \<u> 1473192346 \</u> 1446}\right]")
(51) Type: Record(ev: Vector(Float),EV: Matrix(Float)) fricastestEquals("checkResult(R13,M)<eps", "true") Type: Voidfricas-- Example 4.23
M:=matrix [[1 for i in 1..50] for j in 1..50];Type: Matrix(Integer)fricasR23:=jacobi(M);
Type: Record(ev: Vector(Float),EV: Matrix(Float)) fricastestEquals("checkResult(R23,M)<eps", "true") Type: Voidfricas-- Results/Statistics
fricas)set output algebra on
fricas)version
"FriCAS 1.3.10 compiled at Wed 10 Jan 02:19:45 CET 2024"fricas)lisp (lisp-implementation-type)
Your user access level is compiler and this command is therefore not available. See the )set userlevel command for more information.fricas)lisp (lisp-implementation-version)
Your user access level is compiler and this command is therefore not available. See the )set userlevel command for more information.
statistics()
============================================================================= General WARNINGS: * do not use ')clear completely' before having used 'statistics()' It clears the statistics without warning! * do not forget to pass the arguments of the testXxxx functions as Strings! Otherwise,the test will fail and statistics() will not notice!
============================================================================= Testsuite: Package JacobiDiagonalisation failed (total): 0 (1)
============================================================================= testsuite | testcases: failed (total) | tests: failed (total) Package JacobiDiagonalisation 0 (1) 0 (16) ============================================================================= File summary. unexpected failures: 0 expected failures: 0 unexpected passes: 0 total tests: 16Type: Void