MathAction SandBoxJacobiDiag

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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

spad

   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: 0.03 SEC.

   compiling exported jacobi : Matrix Float -> Record(ev: Vector Float,EV: Matrix Float)
Time: 0.09 SEC.

   compiling exported checkResult : (Record(ev: Vector Float,EV: Matrix Float),Matrix Float) -> Float
Time: 0.02 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: 0.14 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 16 JUL 2018 08:09:39 PM):

; /var/aw/var/LatexWiki/JACDIAG.NRLIB/JACDIAG.fasl written
; compilation finished in 0:00:00.415
------------------------------------------------------------------------
   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

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 jacdiag

fricas

)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

fricas

-- Cases
testcase "jacobi"

   All user variables and function definitions have been cleared.

Type: Void

fricas

-- 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)

fricas

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)

fricas

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))

fricas

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)

fricas

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)

fricas

eps:=0.0000000001

$\label{eq6}0.1 E - 9$

(6)

Type: Float

fricas

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

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

Type: Void

fricas

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)

fricas

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

fricas

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

Type: Void

fricas

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

Type: Void

fricas

-- 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)

fricas

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)

fricas

p:=showParams()

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

(10)

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

fricas

p.maxit:=20000

$\label{eq11}20000$

(11)

Type: PositiveInteger?

fricas

p.tol:=0.00000000001

$\label{eq12}0.1 E - 10$

(12)

Type: Float

fricas

setParams(p)

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

(13)

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

fricas

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="793" height="155"/>

(14)

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

fricas

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

Type: Void

fricas

-- 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)

fricas

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))

fricas

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

Type: Void

fricas

-- 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)

fricas

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)

fricas

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))

fricas

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

Type: Void

fricas

-- 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)

fricas

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))

fricas

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

Type: Void

fricas

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

$\label{eq24}0.1 E - 7$

(24)

Type: Float

fricas

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)

fricas

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))

fricas

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

Type: Void

fricas

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

$\label{eq27}0.1 E - 6$

(27)

Type: Float

fricas

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?)

fricas

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

Value = "FriCAS 1.3.4 compiled at Wed Jun 27 14:53:01 UTC 2018"

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)lisp (lisp-implementation-type)

Value = "SBCL"

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)lisp (lisp-implementation-version)

Value = "1.1.1"

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!
   * testLibraryError does not prevent FriCAS from aborting the current block.
     Thus, if a block contains other test functions, they will not be executed
     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