MathAction SandBoxJacobiDiag

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(1) -> <spad>

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)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,tol: Float)
Time: 0 SEC.

   compiling exported setParams : Record(maxit: PositiveInteger,tol: Float) -> Record(maxit: PositiveInteger,tol: Float)
Time: 0 SEC.

   compiling local maxInd : (PositiveInteger,NonNegativeInteger,Matrix Float) -> PositiveInteger
Time: 0.01 SEC.

   compiling exported jacobi : Matrix Float -> Record(ev: Vector Float,EV: Matrix Float)
Time: 0.04 SEC.

   compiling exported checkResult : (Record(ev: Vector Float,EV: Matrix Float),Matrix Float) -> Float
Time: 0 SEC.

(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/JACDIAG

Test 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

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)set break resume

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)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 jacdiag

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)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: Void

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-- Cases
testcase "jacobi"

   All user variables and function definitions have been cleared.

Type: Void

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-- 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]]

$\label{eq1}\left[ \begin{array}{cccc} 4 & -{30}&{60}& -{35} \ -{30}&{300}& -{675}&{420} \ {60}& -{675}&{1620}& -{1050} \ -{35}&{420}& -{1050}&{700}$

(1)

Type: Matrix(Integer)

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M:=map(s+->s::Float, S)

$\label{eq2}\left[ \begin{array}{cccc} {4.0}& -{30.0}&{60.0}& -{35.0} \ -{30.0}&{300.0}& -{675.0}&{420.0} \ {60.0}& -{675.0}&{1620.0}& -{1050.0} \ -{35.0}&{420.0}& -{1050.0}&{700.0}$

(2)

Type: Matrix(Float)

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R:=jacobi(M)

$\label{eq3}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle ev ={ \begin{array}{@{}l} \displaystyle \left[{0.1666428611 \ 7189046264}, \:{37.1014913651 \ 27658 188}, \: \right. \ \ \displaystyle \left.{2585.2538109289 \ 223144}, \:{1.4780548447 \ 7813691 18}\right]$

(3)

Type: Record(ev: Vector(Float),EV: Matrix(Float))

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ev1:= vector [0.1666428611_7189046264, 37.1014913651_27658188,
             2585.2538109289_223144, 1.4780548447_781369118]

$\label{eq4}\begin{array}{@{}l} \displaystyle \left[{0.1666428611 \ 7189046264}, \:{37.1014913651 \ 27658 188}, \: \right. \ \ \displaystyle \left.{2585.2538109289 \ 223144}, \:{1.4780548447 \ 7813691 18}\right]$

(4)

Type: Vector(Float)

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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]]

$\label{eq5}\left[ \begin{array}{cccc} {0.7926082911 \ 6404284196}& -{0.1791862905 \ 3191911245}&{0.0 291933231 \ 7921032382 \ 1}& -{0.5820756994 \ 9722239026} \ {0.4519231209 \ 0048280218}&{0.7419177906 \ 0266858684}& -{0.3 287120558 \ 2291241902}&{0.3705021850 \ 6710175866} \ {0.3224163985 \ 8196511663}& -{0.1002281368 \ 8300231266}&{0.7 914111458 \ 4119456574}&{0.5095786345 \ 0180583406} \ {0.2521611696 \ 891868645}& -{0.6382825282 \ 3465850686}& -{0.5145527499 \ 457719896}&{0.5140482722 \_ 2216914813}$

(5)

Type: Matrix(Float)

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eps:=0.0000000001

$\label{eq6}0.1 E - 9$

(6)

Type: Float

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floatNull(v:Vector Float):Boolean == sqrt dot(v,v) <= eps 

   Function declaration floatNull : Vector(Float) -> Boolean has been 
      added to workspace.

Type: Void

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vector members(map(abs,R.EV-EV1))

$\label{eq7}\begin{array}{@{}l} \displaystyle \left[{0.0}, \:{0.0}, \:{0.3 E - 21}, \:{0.0}, \:{0.0}, \:{0.3 E - 20}, \:{0.3 E - 20}, \:{0.5 E - 20}, \right. \ \ \displaystyle \left.\:{0.2 E - 20}, \:{0.3 E - 20}, \:{0.0}, \:{0.0}, \:{0.3 E - 20}, \:{0.3 E - 20}, \:{0.0}, \: \right. \ \ \displaystyle \left.{0.3 E - 20}\right]$

(7)

Type: Vector(Float)

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testEquals("floatNull(R.ev-ev1)", "true")

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Compiling function floatNull with type Vector(Float) -> Boolean

Type: Void

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testEquals("floatNull(vector members(map(abs,R.EV-EV1)))", "true")

Type: Void

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-- 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]]

$\label{eq8}\left[ \begin{array}{cccc} 5 & 4 & 1 & 1 \ 4 & 5 & 1 & 1 \ 1 & 1 & 4 & 2 \ 1 & 1 & 2 & 4$

(8)

Type: Matrix(Integer)

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M:=map(s+->s::Float, S)

$\label{eq9}\left[ \begin{array}{cccc} {5.0}&{4.0}&{1.0}&{1.0} \ {4.0}&{5.0}&{1.0}&{1.0} \ {1.0}&{1.0}&{4.0}&{2.0} \ {1.0}&{1.0}&{2.0}&{4.0}$

(9)

Type: Matrix(Float)

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p:=showParams()

$\label{eq10}\left[{maxit ={10000}}, \:{tol ={0.1 E - 8}}\right]$

(10)

Type: Record(maxit: PositiveInteger?,tol: Float)

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p.maxit:=20000

$\label{eq11}20000$

(11)

Type: PositiveInteger?

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p.tol:=0.00000000001

$\label{eq12}0.1 E - 10$

(12)

Type: Float

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setParams(p)

$\label{eq13}\left[{maxit ={20000}}, \:{tol ={0.1 E - 10}}\right]$

(13)

Type: Record(maxit: PositiveInteger?,tol: Float)

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R1:=jacobi(M)

$\label{eq14}\begin{array}{@{}l} \displaystyle \left[{ev ={\left[{1.0}, \:{10.0}, \:{2.0}, \:{5.0}\right]}}, \: \right. \ \ \displaystyle \left.{ \begin{array}{@{}l} \displaystyle EV = \ \ \displaystyle {\left[ \begin{array}{cccc} {0.7071067811 \ 865475244}&{0.6324555320 \ 336758664}&{0.0}& -{0.3162277660 \ 168379332} \ -{0.7071067811 \ 865475244}&{0.6324555320 \ 336758664}&{0.0}& -{0.3162277660 \ 168379332} \ {0.0}&{0.3162277660 \ 168379332}&{0.7071067811 \ 865475244}&{0.6 324555320 \ 336758664} \ {0.0}&{0.3162277660 \ 168379332}& -{0.7071067811 \ 86547524 4}&{0.6324555320 \ 336758664}$ ?}}, \: \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))

fricas

testEquals("checkResult(R1,M)<eps","true")

Type: Void

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-- Example 4.2
S:= matrix [[6,4,4,1], [4,6,1,4], [4,1,6,4],[1,4,4,6]]

$\label{eq15}\left[ \begin{array}{cccc} 6 & 4 & 4 & 1 \ 4 & 6 & 1 & 4 \ 4 & 1 & 6 & 4 \ 1 & 4 & 4 & 6$

(15)

Type: Matrix(Integer)

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M:=map(s+->s::Float, S)

$\label{eq16}\left[ \begin{array}{cccc} {6.0}&{4.0}&{4.0}&{1.0} \ {4.0}&{6.0}&{1.0}&{4.0} \ {4.0}&{1.0}&{6.0}&{4.0} \ {1.0}&{4.0}&{4.0}&{6.0}$

(16)

Type: Matrix(Float)

fricas

R2:=jacobi(M)

$\label{eq17}\begin{array}{@{}l} \displaystyle \left[{ev ={\left[ -{1.0}, \:{5.0}, \:{5.0}, \:{15.0}\right]}}, \: \right. \ \ \displaystyle \left.{ \begin{array}{@{}l} \displaystyle EV = \ \ \displaystyle {\left[ \begin{array}{cccc} {0.5}&{0.5}&{0.5}&{0.5} \ -{0.5}&{0.5}& -{0.5}&{0.5} \ -{0.5}& -{0.5}&{0.5}&{0.5} \ {0.5}& -{0.5}& -{0.5}&{0.5}$

(17)

Type: Record(ev: Vector(Float),EV: Matrix(Float))

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testEquals("checkResult(R2,M)<eps","true")

Type: Void

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-- Example 4.3
S:= matrix [[2,1,3,4], [1,-3,1,5], [3,1,6,-2],[4,5,-2,-1]]

$\label{eq18}\left[ \begin{array}{cccc} 2 & 1 & 3 & 4 \ 1 & - 3 & 1 & 5 \ 3 & 1 & 6 & - 2 \ 4 & 5 & - 2 & - 1$

(18)

Type: Matrix(Integer)

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M:=map(s+->s::Float, S)

$\label{eq19}\left[ \begin{array}{cccc} {2.0}&{1.0}&{3.0}&{4.0} \ {1.0}& -{3.0}&{1.0}&{5.0} \ {3.0}&{1.0}&{6.0}& -{2.0} \ {4.0}&{5.0}& -{2.0}& -{1.0}$

(19)

Type: Matrix(Float)

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R3:=jacobi(M)

$\label{eq20}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle ev ={ \begin{array}{@{}l} \displaystyle \left[ -{1.5731907383 \ 035074402}, \: -{8.0285783523 \ 965 302993}, \right. \ \ \displaystyle \left.\:{7.9329047178 \ 700173785}, \:{5.6688643728 \ 30020 3611}\right]$

(20)

Type: Record(ev: Vector(Float),EV: Matrix(Float))

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testEquals("checkResult(R3,M)<eps","true")

Type: Void

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-- Example 4.4
S:= matrix [[5,7,6,5], [7,10,8,7], [6,8,10,9],[5,7,9,10]]

$\label{eq21}\left[ \begin{array}{cccc} 5 & 7 & 6 & 5 \ 7 &{10}& 8 & 7 \ 6 & 8 &{10}& 9 \ 5 & 7 & 9 &{10}$

(21)

Type: Matrix(Integer)

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M:=map(s+->s::Float, S)

$\label{eq22}\left[ \begin{array}{cccc} {5.0}&{7.0}&{6.0}&{5.0} \ {7.0}&{10.0}&{8.0}&{7.0} \ {6.0}&{8.0}&{10.0}&{9.0} \ {5.0}&{7.0}&{9.0}&{10.0}$

(22)

Type: Matrix(Float)

fricas

R4:=jacobi(M)

$\label{eq23}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle ev ={ \begin{array}{@{}l} \displaystyle \left[{0.0101500483 \ 9789186807 \ 2}, \:{3.8580574559 \_ 4 49508547}, \right. \ \ \displaystyle \left.\:{0.8431071498 \ 550318408}, \:{30.2886853458 \ 0212 5436}\right]$

(23)

Type: Record(ev: Vector(Float),EV: Matrix(Float))

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testEquals("checkResult(R4,M)<eps","true")

Type: Void

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-- Example 4.5
eps:=0.00000001  -- eps from above too small

$\label{eq24}0.1 E - 7$

(24)

Type: Float

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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]]

$\label{eq25}\left[ \begin{array}{ccccc} {0.81321}& -{0.00013}&{0.00014}&{0.00011}&{0.00021} \ -{0.00013}&{0.93125}&{0.23567}&{0.41235}&{0.41632} \ {0.00014}&{0.23567}&{0.18765}&{0.50632}&{0.30697} \ {0.00011}&{0.41235}&{0.50632}&{0.27605}&{0.46322} \ {0.00021}&{0.41632}&{0.30697}&{0.46322}&{0.41931}$

(25)

Type: Matrix(Float)

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R5:=jacobi(M)

$\label{eq26}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle ev ={ \begin{array}{@{}l} \displaystyle \left[{0.8132101714 \ 8644666656}, \:{0.4198520858 \ 425381 4826}, \: \right. \ \ \displaystyle \left.-{0.2990822128 \_ 7531886141}, \: \right. \ \ \displaystyle \left.{1.6782798304 \ 035586382}, \:{0.0152101251 \ 4277540 844 \_ 1}\right]$

(26)

Type: Record(ev: Vector(Float),EV: Matrix(Float))

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testEquals("checkResult(R5,M)<eps","true")

Type: Void

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-- Example 4.6
eps:=0.0000001  -- eps from above too small

$\label{eq27}0.1 E - 6$

(27)

Type: Float

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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]]

$\label{eq28}\left[ \begin{array}{ccccc} 5 & 4 & 3 & 2 & 1 \ 4 & 6 & 0 & 4 & 3 \ 3 & 0 & 7 & 6 & 5 \ 2 & 4 & 6 & 8 & 7 \ 1 & 3 & 5 & 7 & 9$

(28)

Type: Matrix(NonNegativeInteger?)

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M:=map(s+->s::Float, S)

$\label{eq29}\left[ \begin{array}{ccccc} {5.0}&{4.0}&{3.0}&{2.0}&{1.0} \ {4.0}&{6.0}&{0.0}&{4.0}&{3.0} \ {3.0}&{0.0}&{7.0}&{6.0}&{5.0} \ {2.0}&{4.0}&{6.0}&{8.0}&{7.0} \ {1.0}&{3.0}&{5.0}&{7.0}&{9.0}$

(29)

Type: Matrix(Float)

fricas

R6:=jacobi(M)

$\label{eq30}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle ev ={ \begin{array}{@{}l} \displaystyle \left[ -{1.0965951816 \ 58696809}, \:{7.5137241542 \ 053727 573}, \: \right. \ \ \displaystyle \left.{4.8489501203 \ 161481508}, \:{1.3270455995 \ 5676524 47}, \: \right. \ \ \displaystyle \left.{22.4068753075 \_ 80410656}\right]$

(30)

Type: Record(ev: Vector(Float),EV: Matrix(Float))

fricas

testEquals("checkResult(R6,M)<eps","true")

Type: Void

fricas

-- 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]]

$\label{eq31}\left[ \begin{array}{ccccc} {10}& 1 & 2 & 3 & 4 \ 1 & 9 & - 1 & 2 & - 3 \ 2 & - 1 & 7 & 3 & - 5 \ 3 & 2 & 3 &{12}& - 1 \ 4 & - 3 & - 5 & - 1 &{15}$

(31)

Type: Matrix(Integer)

fricas

M:=map(s+->s::Float, S)

$\label{eq32}\left[ \begin{array}{ccccc} {10.0}&{1.0}&{2.0}&{3.0}&{4.0} \ {1.0}&{9.0}& -{1.0}&{2.0}& -{3.0} \ {2.0}& -{1.0}&{7.0}&{3.0}& -{5.0} \ {3.0}&{2.0}&{3.0}&{12.0}& -{1.0} \ {4.0}& -{3.0}& -{5.0}& -{1.0}&{15.0}$

(32)

Type: Matrix(Float)

fricas

R7:=jacobi(M)

$\label{eq33}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle ev ={ \begin{array}{@{}l} \displaystyle \left[{6.9948378304 \ 964728165}, \:{9.3655549201 \ 0613240 88}, \: \right. \ \ \displaystyle \left.{1.6552662077 \ 271664822}, \:{15.8089207643 \ 904919 67}, \: \right. \ \ \displaystyle \left.{19.1754202772 \_ 79736326}\right]$

(33)

Type: Record(ev: Vector(Float),EV: Matrix(Float))

fricas

testEquals("checkResult(R7,M)<eps","true")

Type: Void

fricas

-- 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]]

$\label{eq34}\left[ \begin{array}{cccccc} 5 & 1 & - 2 & 0 & - 2 & 5 \ 1 & 6 & - 3 & 2 & 0 & 6 \ - 2 & - 3 & 8 & - 5 & - 6 & 0 \ 0 & 2 & - 5 & 5 & 1 & - 2 \ - 2 & 0 & - 6 & 1 & 6 & - 3 \ 5 & 6 & 0 & - 2 & - 3 & 8$

(34)

Type: Matrix(Integer)

fricas

M:=map(s+->s::Float, S)

$\label{eq35}\left[ \begin{array}{cccccc} {5.0}&{1.0}& -{2.0}&{0.0}& -{2.0}&{5.0} \ {1.0}&{6.0}& -{3.0}&{2.0}&{0.0}&{6.0} \ -{2.0}& -{3.0}&{8.0}& -{5.0}& -{6.0}&{0.0} \ {0.0}&{2.0}& -{5.0}&{5.0}&{1.0}& -{2.0} \ -{2.0}&{0.0}& -{6.0}&{1.0}&{6.0}& -{3.0} \ {5.0}&{6.0}&{0.0}& -{2.0}& -{3.0}&{8.0}$

(35)

Type: Matrix(Float)

fricas

R8:=jacobi(M)

$\label{eq36}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle ev ={ \begin{array}{@{}l} \displaystyle \left[{4.4559896384 \ 593662322}, \: -{1.5987342935 \ 81359 4058}, \right. \ \ \displaystyle \left.\:{16.1427446551 \ 21993174}, \:{4.4559896384 \ 59366 2321}, \: \right. \ \ \displaystyle \left.-{1.5987342935 \ 813594058}, \:{16.1427446551 \ 21993 174}\right]$

(36)

Type: Record(ev: Vector(Float),EV: Matrix(Float))

fricas

testEquals("checkResult(R8,M)<eps","true")

Type: Void

fricas

-- 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]]

$\label{eq37}\left[ \begin{array}{cccccc} 1 & 2 & 3 & 0 & 1 & 2 \ 2 & 4 & 5 & - 1 & 0 & 3 \ 3 & 5 & 6 & - 2 & - 3 & 0 \ 0 & - 1 & - 2 & 1 & 2 & 3 \ 1 & 0 & - 3 & 2 & 4 & 5 \ 2 & 3 & 0 & 3 & 5 & 6$

(37)

Type: Matrix(Integer)

fricas

M:=map(s+->s::Float, S)

$\label{eq38}\left[ \begin{array}{cccccc} {1.0}&{2.0}&{3.0}&{0.0}&{1.0}&{2.0} \ {2.0}&{4.0}&{5.0}& -{1.0}&{0.0}&{3.0} \ {3.0}&{5.0}&{6.0}& -{2.0}& -{3.0}&{0.0} \ {0.0}& -{1.0}& -{2.0}&{1.0}&{2.0}&{3.0} \ {1.0}&{0.0}& -{3.0}&{2.0}&{4.0}&{5.0} \ {2.0}&{3.0}&{0.0}&{3.0}&{5.0}&{6.0}$

(38)

Type: Matrix(Float)

fricas

R9:=jacobi(M)

$\label{eq39}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle ev ={ \begin{array}{@{}l} \displaystyle \left[{0.2849864366 \ 7450951293}, \: -{1.6963228483 \ 9591 36551}, \right. \ \ \displaystyle \left.\:{12.4113364117 \ 21404142}, \:{0.2849864366 \ 74509 51293}, \: \right. \ \ \displaystyle \left.-{1.6963228483 \ 959136551}, \:{12.4113364117 \ 21404 142}\right]$

(39)

Type: Record(ev: Vector(Float),EV: Matrix(Float))

fricas

testEquals("checkResult(R9,M)<eps","true")

Type: Void

fricas

-- Example 4.10
eps:=0.00001  -- eps from above too small

$\label{eq40}0.00001$

(40)

Type: Float

fricas

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]]

$\label{eq41}\left[ \begin{array}{cccccccc} {611}&{196}& -{192}&{407}& - 8 & -{52}& -{49}&{29} \ {196}&{899}&{113}& -{192}& -{71}& -{43}& - 8 & -{44} \ -{192}&{113}&{899}&{196}&{61}&{49}& 8 &{52} \ {407}& -{192}&{196}&{611}& 8 &{44}&{59}& -{23} \ - 8 & -{71}&{61}& 8 &{411}& -{599}&{208}&{208} \ -{52}& -{43}&{49}&{44}& -{599}&{411}&{208}&{208} \ -{49}& - 8 & 8 &{59}&{208}&{208}&{99}& -{911} \ {29}& -{44}&{52}& -{23}&{208}&{208}& -{911}&{99}$

(41)

Type: Matrix(Integer)

fricas

M:=map(s+->s::Float, S)

$\label{eq42}\left[ \begin{array}{cccccccc} {611.0}&{196.0}& -{192.0}&{407.0}& -{8.0}& -{52.0}& -{49.0}&{2 9.0} \ {196.0}&{899.0}&{113.0}& -{192.0}& -{71.0}& -{43.0}& -{8.0}& -{44.0} \ -{192.0}&{113.0}&{899.0}&{196.0}&{61.0}&{49.0}&{8.0}&{52.0} \ {407.0}& -{192.0}&{196.0}&{611.0}&{8.0}&{44.0}&{59.0}& -{23.0} \ -{8.0}& -{71.0}&{61.0}&{8.0}&{411.0}& -{599.0}&{208.0}&{208.0} \ -{52.0}& -{43.0}&{49.0}&{44.0}& -{599.0}&{411.0}&{208.0}&{208.0} \ -{49.0}& -{8.0}&{8.0}&{59.0}&{208.0}&{208.0}&{99.0}& -{911.0} \ {29.0}& -{44.0}&{52.0}& -{23.0}&{208.0}&{208.0}& -{911.0}&{99.0}$

(42)

Type: Matrix(Float)

fricas

R10:=jacobi(M)

$\label{eq43}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle ev ={ \begin{array}{@{}l} \displaystyle \left[{0.0980486407 \ 2151706114 \ 7}, \:{999.9999999999 \_ 9999958}, \right. \ \ \displaystyle \left.\:{1000.0}, \:{1020.0490184299 \_ 968238}, \: \right. \ \ \displaystyle \left.{0.3776805208 \_ 266153749 E - 15}, \: \right. \ \ \displaystyle \left.{1019.9999999999 \ 989334}, \: -{1020.0490184299 \ 96 8238}, \right. \ \ \displaystyle \left.\:{1019.9019513592 \_ 795495}\right]$

(43)

Type: Record(ev: Vector(Float),EV: Matrix(Float))

fricas

testEquals("checkResult(R10,M)<eps","true")

Type: Void

fricas

-- Example 4.11
eps:=0.00001  -- eps from above too small

$\label{eq44}0.00001$

(44)

Type: Float

fricas

S:= diagonalMatrix [5,6,6,6,6,6,6,6,6,6,5]

$\label{eq45}\left[ \begin{array}{ccccccccccc} 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5$

(45)

Type: Matrix(Integer)

fricas

for i in 2..11 repeat S(i-1,i):=S(i,i-1):=3

Type: Void

fricas

for i in 3..11 repeat S(i-2,i):=S(i,i-2):=1

Type: Void

fricas

for i in 4..11 repeat S(i-3,i):=S(i,i-3):=1

Type: Void

fricas

S(1,2):=S(2,1):=S(10,11):=S(11,10):=2

$\label{eq46}2$

(46)

Type: PositiveInteger?

fricas

M:=map(s+->s::Float, S)

$\label{eq47}\left[ \begin{array}{ccccccccccc} {5.0}&{2.0}&{1.0}&{1.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0} \ {2.0}&{6.0}&{3.0}&{1.0}&{1.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0} \ {1.0}&{3.0}&{6.0}&{3.0}&{1.0}&{1.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0} \ {1.0}&{1.0}&{3.0}&{6.0}&{3.0}&{1.0}&{1.0}&{0.0}&{0.0}&{0.0}&{0.0} \ {0.0}&{1.0}&{1.0}&{3.0}&{6.0}&{3.0}&{1.0}&{1.0}&{0.0}&{0.0}&{0.0} \ {0.0}&{0.0}&{1.0}&{1.0}&{3.0}&{6.0}&{3.0}&{1.0}&{1.0}&{0.0}&{0.0} \ {0.0}&{0.0}&{0.0}&{1.0}&{1.0}&{3.0}&{6.0}&{3.0}&{1.0}&{1.0}&{0.0} \ {0.0}&{0.0}&{0.0}&{0.0}&{1.0}&{1.0}&{3.0}&{6.0}&{3.0}&{1.0}&{1.0} \ {0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{1.0}&{1.0}&{3.0}&{6.0}&{3.0}&{1.0} \ {0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{1.0}&{1.0}&{3.0}&{6.0}&{2.0} \ {0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{0.0}&{1.0}&{1.0}&{2.0}&{5.0}$

(47)

Type: Matrix(Float)

fricas

R11:=jacobi(M)

$\label{eq48}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle ev ={ \begin{array}{@{}l} \displaystyle \left[{4.1292484841 \ 890931323}, \:{0.5222822874 \ 6137264 404}, \: \right. \ \ \displaystyle \left.{8.8284271247 \ 461905061}, \:{3.9999999999 \ 9999999 97}, \: \right. \ \ \displaystyle \left.{14.9418193276 \ 76381949}, \:{3.1715728752 \ 5380770 16}, \: \right. \ \ \displaystyle \left.{5.9999999999 \ 99999446}, \:{1.8038475772 \ 93370916 8}, \: \right. \ \ \displaystyle \left.{12.1961524227 \ 06631475}, \:{4.4066499006 \ 7315222 96}, \: \right. \ \ \displaystyle \left.{3.9999999999 \_ 999999999}\right]$

(48)

Type: Record(ev: Vector(Float),EV: Matrix(Float))

fricas

testEquals("checkResult(R11,M)<eps","true")

Type: Void

fricas

-- 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]]

$\label{eq49}\left[ \begin{array}{cccccccccccccc} {0.25}&{0.06675}&{0.04}&{0.02475}&{0.0705}&{0.06375}&{0.06925}&{0.0 205}&{0.036}& -{0.01025}& -{0.00175}&{0.0275}&{0.023}&{0.002} \ {0.06675}&{0.25}&{0.104}&{0.07475}&{0.03625}&{0.11675}&{0.110 5}&{0.06225}&{0.051}&{0.0325}&{0.024}&{0.036}&{0.0635}&{0.053} \ {0.04}&{0.104}&{0.25}&{0.14575}&{0.03725}&{0.07175}&{0.078}&{0.1 22}&{0.11275}&{0.09375}&{0.10175}&{0.096}&{0.143}&{0.1155} \ {0.02475}&{0.07475}&{0.14575}&{0.25}&{0.05375}&{0.07}&{0.0522 5}&{0.128}&{0.12475}&{0.1055}&{0.13}&{0.14575}&{0.13975}&{0.1 3375} \ {0.0705}&{0.03625}&{0.03725}&{0.05375}&{0.25}&{0.04575}&{0.05 75}&{0.057}&{0.0505}&{0.01475}&{0.045}&{0.0715}&{0.053}&{0.01 6} \ {0.06375}&{0.11675}&{0.07175}&{0.07}&{0.04575}&{0.25}&{0.0862 5}&{0.088}&{0.0715}&{0.0485}&{0.032}&{0.04475}&{0.033}&{0.045} \ {0.06925}&{0.1105}&{0.078}&{0.05225}&{0.0575}&{0.08625}&{0.25}&{0.0 8725}&{0.0695}&{0.03725}&{0.04025}&{0.043}&{0.04075}&{0.0145} \ {0.0205}&{0.06225}&{0.122}&{0.128}&{0.057}&{0.088}&{0.08725}&{0.2 5}&{0.141}&{0.13275}&{0.1555}&{0.1305}&{0.11825}&{0.09125} \ {0.036}&{0.051}&{0.11275}&{0.12475}&{0.0505}&{0.0715}&{0.0695}&{0.1 41}&{0.25}&{0.07425}&{0.1075}&{0.09175}&{0.10725}&{0.08225} \ -{0.01025}&{0.0325}&{0.09375}&{0.1055}&{0.01475}&{0.0485}&{0.0 3725}&{0.13275}&{0.07425}&{0.25}&{0.155}&{0.09625}&{0.0995}&{0.0 9425} \ -{0.00175}&{0.024}&{0.10175}&{0.13}&{0.045}&{0.032}&{0.04025}&{0.1 555}&{0.1075}&{0.155}&{0.25}&{0.1335}&{0.1485}&{0.1305} \ {0.0275}&{0.036}&{0.096}&{0.14575}&{0.0715}&{0.04475}&{0.043}&{0.1 305}&{0.09175}&{0.09625}&{0.1335}&{0.25}&{0.111}&{0.10075} \ {0.023}&{0.0635}&{0.143}&{0.13975}&{0.053}&{0.033}&{0.04075}&{0.1 1825}&{0.10725}&{0.0995}&{0.1485}&{0.111}&{0.25}&{0.14325} \ {0.002}&{0.053}&{0.1155}&{0.13375}&{0.016}&{0.045}&{0.0145}&{0.0 9125}&{0.08225}&{0.09425}&{0.1305}&{0.10075}&{0.14325}&{0.25}$

(49)

Type: Matrix(Float)

fricas

R12:=jacobi(M)

$\label{eq50}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle ev ={ \begin{array}{@{}l} \displaystyle \left[{0.1663246021 \ 0025823682}, \:{0.1713075617 \ 897417 1399}, \: \right. \ \ \displaystyle \left.{0.0972092161 \ 6203514813 \ 8}, \:{0.1227875231 \_ 8 17131926}, \right. \ \ \displaystyle \left.\:{0.2677332979 \ 5388836645}, \:{0.4627662026 \ 9414 64008}, \: \right. \ \ \displaystyle \left.{0.1032157624 \_ 3104449922}, \: \right. \ \ \displaystyle \left.{0.0643799133 \ 6741279490 \ 3}, \: \right. \ \ \displaystyle \left.{0.1773563338 \_ 2092625442}, \: \right. \ \ \displaystyle \left.{0.0735971188 \ 5341424867 \ 5}, \:{1.3340348369 \_ 5 64426216}, \right. \ \ \displaystyle \left.\:{0.1434228761 \_ 4650515836}, \: \right. \ \ \displaystyle \left.{0.0842252705 \ 6463554604 \ 4}, \: \right. \ \ \displaystyle \left.{0.2316394839 \_ 7783581793}\right]$

(50)

Type: Record(ev: Vector(Float),EV: Matrix(Float))

fricas

testEquals("checkResult(R12,M)<eps","true")

Type: Void

fricas

-- Example 4.13 (Hilbert Matrix)

H(n) == matrix [[1/(i+j-1) for i in 1..n] for j in 1..n]

Type: Void

fricas

M:=H(20);

fricas

Compiling function H with type PositiveInteger -> Matrix(Fraction(
      Integer))

Type: Matrix(Fraction(Integer))

fricas

R13:=jacobi(M)

$\label{eq51}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle ev ={ \begin{array}{@{}l} \displaystyle \left[{1.9071347204 \ 07253103}, \:{0.4870384065 \ 72048867 81}, \: \right. \ \ \displaystyle \left.{0.0008676711 \ 0917149782 \ 705}, \: \right. \ \ \displaystyle \left.{0.0000048305 \ 1004864734 \ 33279}, \: \right. \ \ \displaystyle \left.{0.0089611286 \ 1485648034 \ 98}, \: \right. \ \ \displaystyle \left.{0.1590216371 \_ 077614324 E - 11}, \: \right. \ \ \displaystyle \left.{0.1413948074 \_ 4628595824 E - 7}, \: \right. \ \ \displaystyle \left.{0.3979401165 \_ 5463231616 E - 11}, \: \right. \ \ \displaystyle \left.{0.0755958213 \ 0544095843 \ 9}, \: \right. \ \ \displaystyle \left.{0.1668930345 \_ 4225524645 E - 11}, \: \right. \ \ \displaystyle \left.{0.1872845593 \_ 6715022839 E - 11}, \: \right. \ \ \displaystyle \left.{0.4156604438 \_ 2412458777 E - 11}, \: \right. \ \ \displaystyle \left.{0.6029936660 \_ 8059110817 E - 9}, \: \right. \ \ \displaystyle \left.{0.1279656833 \_ 3254260344 E - 11}, \: \right. \ \ \displaystyle \left.{0.2827652010 \_ 3941708864 E - 6}, \: \right. \ \ \displaystyle \left.{0.1328459298 \_ 5173200512 E - 12}, \: \right. \ \ \displaystyle \left.{0.9892736125 \_ 7865554553 E - 12}, \: \right. \ \ \displaystyle \left.{0.3879726539 \_ 3557307554 E - 11}, \: \right. \ \ \displaystyle \left.{0.3758601579 \_ 8175153646 E - 11}, \: \right. \ \ \displaystyle \left.{0.0000703343 \ 1473192346 \ 1446}\right]$

(51)

Type: Record(ev: Vector(Float),EV: Matrix(Float))

fricas

testEquals("checkResult(R13,M)<eps","true")

Type: Void

fricas

-- Example 4.23

M:=matrix [[1 for i in 1..50] for j in 1..50];

Type: Matrix(Integer)

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R23:=jacobi(M);

Type: Record(ev: Vector(Float),EV: Matrix(Float))

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testEquals("checkResult(R23,M)<eps","true")

Type: Void

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-- Results/Statistics

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)set output algebra on

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)version

"FriCAS 1.3.10 compiled at Wed 10 Jan 02:19:45 CET 2024"

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)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.

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)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: 16

Type: Void