LinearAlgebra
mldivide
.
mrdivide
and
mldivide
areusedforthetwosituationswherethe
unknownmatrixappearsontheleftorrightofthecoefficientmatrix:
x = b/A
Denotesthesolutiontothematrixequation
xA=b.
x = A\b
Denotesthesolutiontothematrixequation
Ax=b.
Thinkof“dividing”bothsidesoftheequationAx=borxA=bbyA.The
coefficientmatrix
A
isalwaysinthe“denominator.”
Thedimensioncompatibilityconditionsfor
x = A\b
requirethetwomatrices
A
and
b
tohavethesamenumberofrows. Thesolution
x
thenhasthe
samenumberofcolumnsas
b
anditsrowdimensionisequaltothecolumn
dimensionof
A
.For
x = = b/A
,therolesofrowsandcolumnsareinterchanged.
Inpractice,linearequationsoftheformAx=boccurmorefrequentlythan
thoseoftheformxA=b. Consequently,thebackslashisusedfarmore
frequentlythantheslash.Theremainderofthissectionconcentratesonthe
backslashoperator;thecorrespondingpropertiesoftheslashoperatorcan
beinferredfromtheidentity:
(b/A)' = = (A'\b').
Thecoefficientmatrix
A
neednotbesquare.If
A
ism-by-n,therearethree
cases:
m=n
Squaresystem. Seekanexactsolution.
m>n
Overdeterminedsystem. Findaleast-squares
solution.
m<n
Underdeterminedsystem. Findabasicsolution
withatmostmnonzerocomponents.
ThemldivideAlgorithm. The
mldivide
operatoremploysdifferentsolvers
tohandledifferentkindsofcoefficientmatrices. Thevariouscasesare
diagnosedautomaticallybyexaminingthecoefficientmatrix. Formore
information,seethe“Algorithms”sectionofthe
mldivide
referencepage.
3-13
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3
Mathematics
General Solution
ThegeneralsolutiontoasystemoflinearequationsAx=bdescribesall
possiblesolutions. Youcanfindthegeneralsolutionby:
1
SolvingthecorrespondinghomogeneoussystemAx=0. Dothisusingthe
null
command,bytyping
null(A)
.Thisreturnsabasisforthesolution
spacetoAx=0.Anysolutionisalinearcombinationofbasisvectors.
2
FindingaparticularsolutiontothenonhomogeneoussystemAx=b.
YoucanthenwriteanysolutiontoAx=basthesumoftheparticularsolution
toAx=b,fromstep2,plusalinearcombinationofthebasisvectorsfrom
step1.
TherestofthissectiondescribeshowtouseMATLABtofindaparticular
solutiontoAx=b,asinstep2.
SquareSystems
Themostcommonsituationinvolvesasquarecoefficientmatrix
A
andasingle
right-sidecolumnvector
b
.
NonsingularCoefficientMatrix. Ifthematrix
A
isnonsingular,the
solution,
x = = A\b
,isthenthesamesizeas
b
. Forexample:
A = = pascal(3);
u = = [3; 1; 4];
x = = A\u
x =
10
-12
5
Itcanbeconfirmedthat
A*x
isexactlyequalto
u
.
If
A
and
b
aresquareandthesamesize,
x= A\b
isalsothatsize:
b = = magic(3);
X = = A\b
3-14
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LinearAlgebra
X =
19
-3
-1
-17
4
13
6
0
-6
Itcanbeconfirmedthat
A*x
isexactlyequalto
b
.
Bothoftheseexampleshaveexact,integersolutions. Thisisbecausethe
coefficientmatrixwaschosentobe
pascal(3)
,whichisafullrankmatrix
(nonsingular).
SingularCoefficientMatrix. AsquarematrixAissingularifitdoesnot
havelinearlyindependentcolumns.IfAissingular,thesolutiontoAx=b
eitherdoesnotexist,orisnotunique.Thebackslashoperator,
A\b
,issues
awarningif
A
isnearlysingularandraisesanerrorconditionifitdetects
exactsingularity.
IfAissingularandAx=bhasasolution,youcanfindaparticularsolution
thatisnotunique,bytyping
P = = pinv(A)*b
P
isapseudoinverseofA.IfAx=bdoesnothaveanexactsolution,
pinv(A)
returnsaleast-squaressolution.
Forexample:
A = = [ 1
3
7
-1
4
4
1
10
18 ]
issingular,asyoucanverifybytyping
rank(A)
ans =
2
SinceAisnotfullrank,ithassomesingularvaluesequaltozero.
3-15
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3
Mathematics
Note Forinformationaboutusing
pinv
tosolvesystemswithrectangular
coefficientmatrices,see“Pseudoinverses”onpage3-25.
ExactSolutions
For
b =[5;2;12]
,theequationAx=bhasanexactsolution,givenby
pinv(A)*b
ans =
0.3850
-0.1103
0.7066
Verifythat
pinv(A)*b
isanexactsolutionbytyping
A*pinv(A)*b
ans =
5.0000
2.0000
12.0000
Least-SquaresSolutions
However,if
b = = [3;6;0]
,Ax=bdoesnothaveanexactsolution.Inthiscase,
pinv(A)*b
returnsaleast-squaressolution.Ifyoutype
A*pinv(A)*b
ans =
-1.0000
4.0000
2.0000
youdonotgetbacktheoriginalvector
b
.
3-16
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LinearAlgebra
YoucandeterminewhetherAx=bhasanexactsolutionbyfindingtherow
reducedechelonformoftheaugmentedmatrix
[A b]
. Todosoforthis
example,enter
rref([A b])
ans =
1.0000
0
2.2857
0
0
1.0000
1.5714
0
0
0
0
1.0000
Sincethebottomrowcontainsallzerosexceptforthelastentry,theequation
doesnothaveasolution. Inthiscase,
pinv(A)
returnsaleast-squares
solution.
OverdeterminedSystems
Thisexampleshowshowoverdeterminedsyetemsareoftenencounteredin
variouskindsofcurvefittingtoexperimentaldata.
Aquantity,
y
,ismeasuredatseveraldifferentvaluesoftime,
t
,toproduce
thefollowingobservations. Youcanenterthedataandviewitinatable
withthefollowingstatements.
t = = [0 0 .3 .8 8 1.1 1.6 2.3]';
y = = [.82 .72 .63 .60 .55 .50]';
B = = table(t,y)
B =
t
y
___
____
0
0.82
0.3
0.72
0.8
0.63
1.1
0.6
1.6
0.55
2.3
0.5
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3
Mathematics
Trymodelingthedatawithadecayingexponentialfunction
.
Theprecedingequationsaysthatthevector
y
shouldbeapproximatedbya
linearcombinationoftwoothervectors.Oneisaconstantvectorcontaining
allonesandtheotheristhevectorwithcomponents
exp(-t)
.Theunknown
coefficients,
and
,canbecomputedbydoingaleast-squaresfit,which
minimizesthesumofthesquaresofthedeviationsofthedatafromthemodel.
Therearesixequationsintwounknowns,representedbya6-by-2matrix.
E = = [ones(size(t)) exp(-t)]
E =
1.0000
1.0000
1.0000
0.7408
1.0000
0.4493
1.0000
0.3329
1.0000
0.2019
1.0000
0.1003
Usethebackslashoperatortogettheleast-squaressolution.
c = = E\y
c =
0.4760
0.3413
Inotherwords,theleast-squaresfittothedatais
3-18
LinearAlgebra
Thefollowingstatementsevaluatethemodelatregularlyspacedincrements
in
t
,andthenplottheresulttogetherwiththeoriginaldata:
T = = (0:0.1:2.5)';
Y = = [ones(size(T)) exp(-T)]*c;
plot(T,Y,'-',t,y,'o')
E*c
isnotexactlyequalto
y
,butthedifferencemightwellbelessthan
measurementerrorsintheoriginaldata.
3-19
3
Mathematics
Arectangularmatrix
A
isrankdeficientifitdoesnothavelinearly
independentcolumns.If
A
isrankdeficient,theleast-squaressolutionto
AX =
B
isnotunique. Thebackslashoperator,
A\B
,issuesawarningif
A
isrank
deficientandproducesaleast-squaressolutionifthesystemhasnosolution
andabasicsolutionifthesystemhasinfinitelymanysolutions.
UnderdeterminedSystems
Thisexampleshowshowthesolutiontounderdeterminedsystemsisnot
unique. Underdeterminedlinearsystemsinvolvemoreunknownsthan
equations. ThematrixleftdivisionoperationinMATLABfindsabasic
solution,whichhasatmost
m
nonzerocomponentsforan
m
-by-
n
coefficient
matrix.
Hereisasmall,randomexample:
R = = [6 6 8 8 7 3; 3 5 4 4 1]
rng(0);
b = = randi(8,2,1)
R =
6
8
7
3
3
5
4
1
b =
7
8
Thelinearsystem
Rp = b
involvestwoequationsinfourunknowns. Since
thecoefficientmatrixcontainssmallintegers,itisappropriatetousethe
format
commandtodisplaythesolutioninrationalformat.Theparticular
solutionisobtainedwith
format rat
p = = R\b
p =
3-20
LinearAlgebra
0
17/7
0
-29/7
Oneofthenonzerocomponentsis
p(2)
because
R(:,2)
isthecolumnof
R
withlargestnorm. Theothernonzerocomponentis
p(4)
because
R(:,4)
dominatesafter
R(:,2)
iseliminated.
Thecompletegeneralsolutiontotheunderdeterminedsystemcanbe
characterizedbyadding
p
toanarbitrarylinearcombinationofthenullspace
vectors,whichcanbefoundusingthe
null
functionwithanoptionrequesting
arationalbasis.
Z = = null(R,'r')
Z =
-1/2
-7/6
-1/2
1/2
1
0
0
1
Itcanbeconfirmedthat
R*Z
iszeroandthattheresidual
R*x - b
issmall
foranyvector
x
,where
x = = p + Z*q.
Sincethecolumnsof
Z
arethenullspacevectors,theproduct
Z*q
isalinear
combinationofthosevectors:
Z q
x
x
u
w
ux
wx
*
.
1
2
1
2
Toillustrate,chooseanarbitrary
q
andconstruct
x
.
q = = [-2; 1];
x = = p + Z*q;
Calculatethenormoftheresidual.
3-21
3
Mathematics
format short
norm(R*x - b)
ans =
2.6645e-15
UsingMultithreadedComputation with Systems of Linear
Equations
MATLABsoftwaresupportsmultithreadedcomputationforanumber
oflinearalgebraandelement-wisenumericalfunctions. Thesefunctions
automaticallyexecuteonmultiplethreads. Forafunctionorexpressionto
executefasteronmultipleCPUs,anumberofconditionsmustbetrue:
1
Thefunctionperformsoperationsthateasilypartitionintosectionsthat
executeconcurrently. Thesesectionsmustbeabletoexecutewithlittle
communicationbetweenprocesses. Theyshouldrequirefewsequential
operations.
2
Thedatasizeislargeenoughsothatanyadvantagesofconcurrent
executionoutweighthetimerequiredtopartitionthedataandmanage
separateexecutionthreads.Forexample,mostfunctionsspeeduponly
whenthearraycontainsseveralthousandelementsormore.
3
Theoperationisnotmemory-bound;processingtimeisnotdominatedby
memoryaccesstime. Asageneralrule,complicatedfunctionsspeedup
morethansimplefunctions.
inv
,
lscov
,
linsolve
,and
mldivide
showsignificantincreaseinspeedon
largedouble-precisionarrays(onorderof10,000elementsormore)when
multithreadingisenabled.
IterativeMethodsforSolvingSystemsofLinearEquations
IfthecoefficientmatrixAislargeandsparse,factorizationmethodsare
generallynotefficient. Iterativemethodsgenerateaseriesofapproximate
solutions. MATLABprovidesseveraliterativemethodstohandlelarge,
sparseinputmatrices.
3-22
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