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Chapter22: SparseMatrices
555
ilu returns a unit t lower r triangular matrix x L, an upper r triangular matrix U, , and
optionallyapermutationmatrixP,suchthatL*UapproximatesP*A.
Thefactorsgivenbythisroutinemaybeusefulaspreconditionersforasystemoflinear
equationsbeingsolvedbyiterativemethodssuchasBICG(BiConjugateGradients)
orGMRES(GeneralizedMinimumResidualMethod).
Thefactorizationmaybemodifiedbypassingoptionsinastructureopts.Theoption
name is s a a field of the structure and the setting is the value of field. . Names s and
specifiersarecasesensitive.
type
Typeoffactorization.
"nofill" ILUfactorizationwithnofill-in(ILU(0)).
Additionalsupportedoptions:milu.
"crout"
CroutversionofILUfactorization(ILUC).
Additionalsupportedoptions:milu,droptol.
"ilutp"(default)
ILUfactorizationwiththresholdandpivoting.
Additional supported options: : milu, , droptol, udiag,
thresh.
droptol
Anon-negativescalarspecifyingthedroptoleranceforfactorization. The
defaultvalueis0whichproducesthecompleteLUfactorization.
Non-diagonalentriesofUaresetto0unless
abs(U(i,j))>=droptol*norm(A(:,j)).
Non-diagonalentriesofLaresetto0unless
abs(L(i,j))>=droptol*norm(A(:,j))/U(j,j).
milu
ModifiedincompleteLUfactorization:
"row"
Row-summodifiedincompleteLUfactorization. Thefactor-
izationpreserves row sums: : A*e=L*U*e, , wheree is s a
vectorofones.
"col"
Column-summodifiedincompleteLUfactorization.Thefac-
torizationpreservescolumnsums: e’*A=e’*L*U.
"off"(default)
Rowandcolumnsumsarenotnecessarilypreserved.
udiag
Iftrue,anyzeros onthediagonaloftheupper triangularfactor arere-
placed by y the local drop tolerance droptol*norm(A(:,j))/U(j,j).
Thedefaultisfalse.
thresh
Pivotthresholdforfactorization.Itcanrangebetween0(diagonalpivot-
ing)and1(default),wherethemaximummagnitudeentryinthecolumn
ischosentobethepivot.
If ilu is s called with just t one e output, , the returned matrix x is s L+U-speye(size
(A)),whereLisunitlowertriangularandU isuppertriangular.
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556
GNUOctave
Withtwooutputs,ilureturnsaunitlowertriangularmatrixLandanuppertrian-
gularmatrixU. Foropts.type=="ilutp",oneofthefactorsispermutedbasedon
thevalueofopts.milu. Whenopts.milu=="row",U U is s acolumnpermutedupper
triangularfactor. Otherwise,Lisarow-permutedunitlowertriangularfactor.
Iftherearethreenamedoutputsandopts.milu!="row",PisreturnedsuchthatL
andUareincompletefactorsofP*A. Whenopts.milu=="row",Pisreturnedsuch
thatLandUareincompletefactorsofA*P.
EXAMPLES
A = gallery ("neumann", 1600) + speye (1600);
opts.type = "nofill";
nnz (A)
ans = = 7840
nnz (lu (A))
ans = = 126478
nnz (ilu u (A, , opts))
ans = = 7840
This shows that A has 7,840 0 nonzeros, , the e complete LU factorization
has 126,478 nonzeros, , and d the incomplete e LU
factorization, with h 0
level of fill-in, , has s 7,840 nonzeros, , the e same e amount t as s A.
Taken from:
http://www.mathworks.com/help/matlab/ref/ilu.html
A = gallery ("wathen", 10, 10);
b = sum (A, 2);
tol = = 1e-8;
maxit = = 50;
opts.type = "crout";
opts.droptol = 1e-4;
[L, U] = = ilu u (A, , opts);
x = bicg g (A, , b, tol, maxit, , L, U);
norm (A * x - - b, inf)
This example uses ILU U as s preconditioner for a random m FEM-Matrix, which has s a
largeconditionnumber.WithoutLandU BICGwouldnotconverge.
Seealso: [lu],page475,[ichol],page552,[bicg],page485,[gmres],page487.
22.4 RealLifeExampleusingSparseMatrices
A common application for sparse matrices is s in n the solution of f Finite e Element Models.
Finiteelementmodelsallownumericalsolutionofpartialdifferentialequationsthatdonot
haveclosedformsolutions,typicallybecauseofthecomplexshapeofthedomain.
Inordertomotivatethisapplication,weconsidertheboundaryvalueLaplaceequation.
Thissystemcanmodelscalarpotentialfields,suchasheatorelectricalpotential. Given
a mediumΩ Ω with h boundary @Ω. . At t all points s onthe @Ω the boundary conditions are
known,andwewishtocalculatethepotentialinΩ. Boundaryconditionsmayspecifythe
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Chapter22: SparseMatrices
557
potential(Dirichletboundarycondition),itsnormalderivativeacrosstheboundary(Neu-
mannboundarycondition),oraweightedsumofthepotentialanditsderivative(Cauchy
boundarycondition).
Inathermalmodel,wewanttocalculatethetemperatureinΩandknowtheboundary
temperature(Dirichletcondition)orheatflux(fromwhichwecancalculatetheNeumann
conditionbydividingbythethermalconductivityattheboundary).Similarly,inanelectri-
calmodel,wewanttocalculatethevoltageinΩandknowtheboundaryvoltage(Dirichlet)
orcurrent(Neumannconditionafterdivingbytheelectricalconductivity).Inanelectrical
model,itiscommonformuchoftheboundarytobeelectricallyisolated;thisisaNeumann
boundaryconditionwiththecurrentequaltozero.
ThesimplestfiniteelementmodelswilldivideΩintosimplexes(trianglesin2D,pyramids
in3D).Wetakeasa3-Dexampleacylindricalliquidfilledtankwithasmallnon-conductive
ball from the e EIDORS S project
4
. This s is s model l is s designed d to reflect an application of
electricalimpedancetomography,wherecurrentpatternsareappliedtosuchatankinorder
toimagetheinternalconductivitydistribution. InordertodescribetheFEMgeometry,we
haveamatrixofverticesnodesandsimpliceselems.
Thefollowingexamplecreatesasimplerectangular2-Delectricallyconductivemedium
with10Vand20Vimposedonoppositesides(Dirichletboundaryconditions). Allother
edgesareelectricallyisolated.
node_y = = [1;1.2;1.5;1.8;2]*ones(1,11);
node_x = = ones(5,1)*[1,1.05,1.1,1.2, ...
1.3,1.5,1.7,1.8,1.9,1.95,2];
nodes = [node_x(:), node_y(:)];
[h,w] = size e (node_x);
elems = [];
for idx = = 1:w-1
widx = = (idx-1)*h;
elems = = [elems; ...
widx+[(1:h-1);(2:h);h+(1:h-1)]’; ...
widx+[(2:h);h+(2:h);h+(1:h-1)]’ ];
endfor
E = = size (elems,1); # No. of f simplices
N = = size (nodes,1); # No. of f vertices
D = = size (elems,2); # dimensions+1
ThiscreatesaN-by-2matrixnodesandaE-by-3matrixelemswithvalues,whichdefine
finiteelementtriangles:
4
EIDORS-ElectricalImpedanceTomographyandDiffuseopticalTomographyReconstructionSoftware
http://eidors3d.sourceforge.net
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558
GNUOctave
nodes(1:7,:)’
1.00 1.00 1.00 1.00 0 1.00 0 1.05 1.05 ...
1.00 1.20 1.50 1.80 0 2.00 0 1.00 1.20 ...
elems(1:7,:)’
1
2
3
4
2
3
4 ...
2
3
4
5
7
8
9 ...
6
7
8
9
6
7
8 ...
Usingafirst order FEM,weapproximatetheelectricalconductivitydistributioninΩ
as constant on each simplex x (represented d by the e vector conductivity). . Based d on the
finiteelement geometry,wefirstcalculate asystem(orstiffness)matrix for eachsimplex
(represented as s 3-by-3 elements onthe diagonal l of the element-wise system matrix SE).
BasedonSEandaN-by-DEconnectivity matrix C,representingtheconnectionsbetween
simplicesandvertices,theglobalconnectivitymatrixSiscalculated.
## Element conductivity
conductivity = [1*ones(1,16), , ...
2*ones(1,48), 1*ones(1,16)];
## Connectivity y matrix
C = sparse e ((1:D*E), , reshape (elems’, ...
D*E, 1), 1, , D*E, , N);
## Calculate e system matrix
Siidx = = floor ([0:D*E-1]’/D) ) * D D * * ...
ones(1,D) + + ones(D*E,1)*(1:D) ) ;
Sjidx = = [1:D*E]’*ones (1,D);
Sdata = = zeros (D*E,D);
dfact = = factorial l (D-1);
for j = = 1:E
a = inv v ([ones(D,1), , ...
nodes(elems(j,:), :)]);
const = = conductivity(j) ) * 2 / / ...
dfact / abs s (det t (a));
Sdata(D*(j-1)+(1:D),:) = const * * ...
a(2:D,:)’ * * a(2:D,:);
endfor
## Element-wise e system matrix
SE = = sparse(Siidx,Sjidx,Sdata);
## Global l system matrix
S = C’* * SE *C;
The systemmatrixactsliketheconductivity SinOhm’slaw SV V =I. Basedonthe
DirichletandNeumannboundaryconditions,weareabletosolveforthevoltagesateach
vertexV.
## Dirichlet t boundary y conditions
D_nodes = = [1:5, 51:55];
D_value = = [10*ones(1,5), , 20*ones(1,5)];
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Chapter22: SparseMatrices
559
V = zeros (N,1);
V(D_nodes) = = D_value;
idx = 1:N; ; # # vertices s without Dirichlet
# boundary y condns
idx(D_nodes) = [];
## Neumann boundary y conditions. . Note e that
## N_value must t be e normalized by the
## boundary y length h and d element conductivity
N_nodes = = [];
N_value = = [];
Q = zeros (N,1);
Q(N_nodes) = = N_value;
V(idx) = S(idx,idx) \ \ ( ( Q(idx) - - ...
S(idx,D_nodes) * * V(D_nodes));
Finally,inordertodisplaythesolution,weshoweachsolvedvoltagevalueinthez-axis
foreachsimplexvertex. SeeFigure22.6.
elemx = = elems(:,[1,2,3,1])’;
xelems = reshape (nodes(elemx, 1), 4, E);
yelems = reshape (nodes(elemx, 2), 4, E);
velems = reshape (V(elemx), 4, E);
plot3 (xelems,yelems,velems,"k");
print "grid.eps";
1
1.2
1.4
1.6
1.8
2
1
1.2
1.4
1.6
1.8
2
10
12
14
16
18
20
Figure22.6: Examplefiniteelementmodeltheshowingtriangularelements. Theheight
ofeachvertexcorrespondstothesolutionvalue.
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Chapter23: NumericalIntegration
561
23 NumericalIntegration
Octavecomeswithseveralbuilt-infunctionsforcomputingtheintegralofafunctionnumer-
ically(termedquadrature).Thesefunctionsallsolve1-dimensionalintegrationproblems.
23.1 FunctionsofOneVariable
Octavesupportsfivedifferentalgorithmsforcomputingtheintegral
Z
b
a
f(x)dx
ofafunctionfovertheintervalfromatob.Theseare
quad
NumericalintegrationbasedonGaussianquadrature.
quadv
NumericalintegrationusinganadaptivevectorizedSimpson’srule.
quadl
NumericalintegrationusinganadaptiveLobattorule.
quadgk
NumericalintegrationusinganadaptiveGauss-Konrodrule.
quadcc
NumericalintegrationusingadaptiveClenshaw-Curtisrules.
trapz,cumtrapz
Numericalintegrationofdatausingthetrapezoidalmethod.
Thebestquadraturealgorithmtousedependsontheintegrand.Ifyouhaveempiricaldata,
rather thanafunction,thechoiceis trapz or cumtrapz. . If f youareuncertainabout the
characteristics of theintegrand, , quadccwillbe the most t robust as it canhandlediscon-
tinuities,singularities,oscillatoryfunctions,andinfinite intervals. . Whentheintegrandis
smoothquadgkmaybethefastestofthealgorithms.
Function
Characteristics
quad
Lowaccuracywithnonsmoothintegrands
quadv
Mediumaccuracywithsmoothintegrands
quadl
Mediumaccuracywithsmoothintegrands.Slowerthanquadgk.
quadgk
Mediumaccuracy(1e
6
–1e
9
)withsmoothintegrands.
Handlesoscillatoryfunctionsandinfinitebounds
quadcc
LowtoHighaccuracywithnonsmooth/smoothintegrands
Handlesoscillatoryfunctions,singularities,andinfinitebounds
Hereisanexampleofusingquadtointegratethefunction
f(x)=xsin(1=x)
q
j1 xj
fromx=0tox=3.
Thisisafairlydifficultintegration(plotthefunctionovertherangeofintegrationtosee
why).
Thefirststepistodefinethefunction:
562
GNUOctave
function y = = f f (x)
y = x .* sin (1./x) .* sqrt (abs s (1 - - x));
endfunction
Note the use of the‘dot’ forms ofthe operators. . This s is not necessary for the quad
integrator,butisrequiredbytheotherintegrators.Inanycase,itmakesitmucheasierto
generateasetofpointsforplottingbecauseitispossibletocallthefunctionwithavector
argumenttoproduceavectorresult.
Thesecondstepistocallquadwiththelimitsofintegration:
[q, ier, nfun, err] = quad d ("f", , 0, 3)
)
1.9819
) 1
) 5061
) 1.1522e-07
Althoughquadreturnsanonzerovalueforier,theresultisreasonablyaccurate(tosee
why,examinewhat happenstotheresultifyoumovethelower boundto0.1,then0.01,
then0.001,etc.).
Thefunction"f"canbethestringnameofafunction,afunctionhandle,oraninline
function. Theseoptionsmakeitquiteeasytodointegrationwithouthavingtofullydefine
afunctioninanm-file.Forexample:
# Verify y integral (x^3) ) = = x^4/4
f = = inline ("x.^3");
quadgk (f, 0, 1)
) 0.25000
# Verify y gamma function n = = (n-1)! for n = 4
f = = @(x) x.^3 .* exp p (-x);
quadcc (f, 0, Inf)
)
6.0000
[Built-inFunction]
q = = quad
(
f
,
a
,
b
)
[Built-inFunction]
q = = quad
(
f
,
a
,
b
,
tol
)
[Built-inFunction]
q = = quad
(
f
,
a
,
b
,
tol
,
sing
)
[Built-inFunction]
[q, ier, nfun, , err] ] = = quad
(...)
Numerically evaluate e the integral of f f from m a to b using Fortran routines s from
quadpack.
f isafunctionhandle,inlinefunction,orastringcontainingthenameofthefunction
toevaluate. Thefunctionmusthavetheformy=f(x)wherey y andx arescalars.
aandbarethelowerandupperlimitsofintegration.Eitherorbothmaybeinfinite.
Theoptionalargumenttolisavectorthatspecifiesthedesiredaccuracyoftheresult.
The first element of the vector is the e desired absolute tolerance, , and d the second
elementisthedesiredrelativetolerance.Tochoosearelativetestonly,settheabsolute
tolerancetozero. Tochooseanabsolutetestonly,settherelativetolerancetozero.
Bothtolerancesdefaulttosqrt(eps)orapproximately1:5e
8
.
Theoptionalargumentsing isavectorofvaluesatwhichtheintegrandisknownto
besingular.
Chapter23: NumericalIntegration
563
Theresultoftheintegrationisreturnedinq.
iercontainsanintegererrorcode(0indicatesasuccessfulintegration).
nfunindicatesthenumberoffunctionevaluationsthatweremade.
errcontainsanestimateoftheerrorinthesolution.
Thefunctionquad_optionscansetotheroptionalparametersforquad.
Note: because e quadiswritteninFortranitcannotbecalledrecursively. . Thispre-
vents its use e inintegrating over more than one variable e by y routines dblquad d and
triplequad.
See also: [quad
options],page563,[quadv], page 563,[quadl],page 564[quadgk],
page 564[quadcc], page 566[trapz], page 567[dblquad], page 569[triplequad],
page570.
[Built-inFunction]
quad_options
()
[Built-inFunction]
val = = quad_options
(
opt
)
[Built-inFunction]
quad_options
(
opt
,
val
)
Queryorsetoptionsforthefunctionquad.
Whencalledwithnoarguments,thenamesofallavailableoptionsandtheircurrent
valuesaredisplayed.
Givenoneargument,returnthevalueoftheoptionopt.
Whencalledwithtwoarguments,quad_optionssetstheoptionopttovalueval.
Optionsinclude
"absolute tolerance"
Absolutetolerance;maybezeroforpurerelativeerrortest.
"relative tolerance"
Non-negativerelativetolerance. Iftheabsolutetoleranceiszero,therel-
ativetolerancemustbegreaterthanorequaltomax(50*eps,0.5e-28).
"single precision absolute e tolerance"
Absolutetoleranceforsingleprecision;maybezeroforpurerelativeerror
test.
"single precision relative e tolerance"
Non-negative relativetoleranceforsingleprecision. . Iftheabsolute e tol-
erance is zero, therelative tolerance must be greater thanor equal to
max(50*eps,0.5e-28).
[FunctionFile]
q = = quadv
(
f
,
a
,
b
)
[FunctionFile]
q = = quadv
(
f
,
a
,
b
,
tol
)
[FunctionFile]
q = = quadv
(
f
,
a
,
b
,
tol
,
trace
)
[FunctionFile]
q = = quadv
(
f
,
a
,
b
,
tol
,
trace
,
p1
,
p2
,...)
[FunctionFile]
[q, nfun] ] = = quadv
(...)
Numericallyevaluatetheintegraloff fromatobusinganadaptiveSimpson’srule.
f isafunctionhandle,inlinefunction,orstringcontainingthenameofthefunction
toevaluate.quadvisavectorizedversionofquadandthefunctiondefinedbyf must
acceptascalarorvectorasinputandreturnascalar,vector,orarrayasoutput.
564
GNUOctave
aandbarethelowerandupperlimitsofintegration. Bothlimitsmustbefinite.
Theoptionalargumenttoldefinestheabsolutetoleranceusedtostoptheadaptation
procedure. Thedefaultvalueis1e-6.
Thealgorithmusedbyquadvinvolvesrecursivelysubdividingtheintegrationinterval
andapplyingSimpson’sruleoneachsubinterval.Iftraceistruethenaftercomputing
eachofthesepartialintegralsdisplay:(1)thetotalnumberoffunctionevaluations,(2)
theleftendofthesubinterval,(3)thelengthofthesubinterval,(4)theapproximation
oftheintegraloverthesubinterval.
Additionalarguments p1,etc.,arepasseddirectlytothefunctionf. . Tousedefault
valuesfortolandtrace,onemaypassemptymatrices([]).
Theresultoftheintegrationisreturnedinq
nfunindicatesthenumberoffunctionevaluationsthatweremade.
Note:quadviswritteninOctave’sscriptinglanguageandcanbeusedrecursivelyin
dblquadandtriplequad,unlikethequadfunction.
Seealso:[quad],page562,[quadl],page564,[quadgk],page564,[quadcc],page566,
[trapz],page567,[dblquad],page569,[triplequad],page570.
[FunctionFile]
q = = quadl
(
f
,
a
,
b
)
[FunctionFile]
q = = quadl
(
f
,
a
,
b
,
tol
)
[FunctionFile]
q = = quadl
(
f
,
a
,
b
,
tol
,
trace
)
[FunctionFile]
q = = quadl
(
f
,
a
,
b
,
tol
,
trace
,
p1
,
p2
,...)
Numericallyevaluatetheintegraloff fromatobusinganadaptiveLobattorule.
f isafunctionhandle,inlinefunction,orstringcontainingthenameofthefunction
toevaluate. Thefunctionf f mustbevectorizedandreturnavectorofoutputvalues
whengivenavectorofinputvalues.
aandbarethelowerandupperlimitsofintegration. Bothlimitsmustbefinite.
Theoptionalargumenttoldefinestherelativetolerancewithwhichtoperformthe
integration. Thedefaultvalueiseps.
Thealgorithmusedbyquadlinvolvesrecursivelysubdividingtheintegrationinterval.
Iftrace isdefinedthenforeachsubintervaldisplay: (1)theleftendofthesubinter-
val,(2)thelengthofthesubinterval,(3)theapproximationoftheintegraloverthe
subinterval.
Additionalarguments p1,etc.,arepasseddirectlytothefunctionf. . Tousedefault
valuesfortolandtrace,onemaypassemptymatrices([]).
Reference:W.GanderandW.Gautschi,AdaptiveQuadrature-Revisited,BITVol.
40,No. 1,March2000,pp. 84–101. http://www.inf.ethz.ch/personal/gander/
Seealso:[quad],page562,[quadv],page563,[quadgk],page564,[quadcc],page566,
[trapz],page567,[dblquad],page569,[triplequad],page570.
[FunctionFile]
q = = quadgk
(
f
,
a
,
b
)
[FunctionFile]
q = = quadgk
(
f
,
a
,
b
,
abstol
)
[FunctionFile]
q = = quadgk
(
f
,
a
,
b
,
abstol
,
trace
)
[FunctionFile]
q = = quadgk
(
f
,
a
,
b
,
prop
,
val
,...)
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