Numerical method1 Iterative approaches for solving linear systems Successive over-relaxation Post-nonlinear system Partial pivoting Lecture 11-II

Numerical methodNumerical method 11

Iterative approaches for solving linear Iterative approaches for solving linear systemssystems

Successive over-relaxationSuccessive over-relaxationPost-nonlinear systemPost-nonlinear systemPartial pivotingPartial pivoting

Lecture 11-IILecture 11-II


Derivation ofDerivation of Successive Successive over-relaxationover-relaxation


Successive over-relaxationSuccessive over-relaxation methodmethod

D=diag(diag(A));U=triu(A)-D;L=A-triu(A);R=D+w*L;T=inv(R)*((1-w)*D-w*U);c=w*inv(R)*b;

wbxwUDwwLDx ))1(()( 1

wbxwUDwxwLD ))1(()(


Successive over-relaxationSuccessive over-relaxation methodmethod

A,b it_xor.mx

A=[10 -1 2 0;-1 11 -1 3; 2 -1 10 -1;0 3 -1 8];b=[6 25 -11 15]';inv(A)*b


>> it_xor(A,b) It takes 17 iterations to converge

ans =

1.00000000300255 2.00000000357295 -1.00000000163021 0.99999999724547


VariantVariant successive over-successive over-relaxationrelaxation

bxUwDxLDwD )()(


Variant Successive over-Variant Successive over-relaxationrelaxation

D=diag(diag(A));U=triu(A)-D;L=A-triu(A);R=w*D+D+L;T=inv(R)*(w*D-U);c=inv(R)*b;

bxUwDxLDwD )()(

bLDwDc

UwDLDwD1

1

)(

)()(T


Variant Successive over-Variant Successive over-relaxationrelaxation method method

A,b it_vxor.mx

A=[10 -1 2 0;-1 11 -1 3; 2 -1 10 -1;0 3 -1 8];b=[6 25 -11 15]';inv(A)*b


>> it_vxor(A,b,w) It takes 17 iterations to converge

ans =

1.00000000300255 2.00000000357295 -1.00000000163021 0.99999999724547


w=0w=0

>> it_vxor(A,b,0) It takes 10 iterations to converge

ans =

0.99999999998681 1.99999999985957 -0.99999999997639 1.00000000005561


Large wLarge w

>> it_vxor(A,b,3) It takes 90 iterations to converge

ans =

0.99999998371358 1.99999997234472 -0.99999998858309 1.00000003116601


Post-Nonlinear SystemPost-Nonlinear System

Ax=b, Ax=b, A=rand(100,100),b=rand(100,1)A=rand(100,100),b=rand(100,1)

f(Ax)=b, where f is an arbitrary one-f(Ax)=b, where f is an arbitrary one-dimensional nonlinear functiondimensional nonlinear function

Given A,b Given A,b

find f and xfind f and x


f(Ax)=bf(Ax)=b

Multiple solutionsMultiple solutionsx

fmean absolute error: 0.017080mean square error: 0.000755

mean(abs(b-f(Ax)))


f(Ax)=bf(Ax)=b

Multiple solutionsMultiple solutionsx f

mean absolute error: 0.024637mean square error: 0.002738


Matrix inversionMatrix inversion

Solve Gx=fSolve Gx=f ExampleExample

Gf.zip

>> load Gf.mat;>> inv(G);Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 4.206041e-017.


rankrank

ss=cputime;for i=1:1000rank(G);endss2=cputime;fprintf('cputime: rank %f \n',ss2-ss);cputime: rank 1.703125


rankrank

ss=cputime;for i=1:50000rank(G);endss2=cputime;fprintf('cputime: rank %f \n',ss2-ss);cputime: rank 82.546875


pinvpinv

ss=cputime;for i=1:1000pinv(G);endss2=cputime;fprintf('cputime: pinv %f \n',ss2-ss);

cputime: pinv 7.062500


Inverse of submatrixInverse of submatrix

ss=cputime;for i=1:50000inv(G(1:61,1:61));endss2=cputime;fprintf('cputime: pinv %f \n',ss2-ss);cputime: inv of submatrix 28.937500

Time reduced from 50*7.06 to 28.93 sec


Partial pivoting Partial pivoting

Maximal column pivotingMaximal column pivoting

78.46130.629.5 :

17.5917.5916.^10:

212

211

xxE

xxE

The pivot or pivot element is the element of a matrix, which is selected first by an algorithm (e.g. Gaussian elimination, Quicksort, Simplex algorithm), to do certain calculations with the matrix

http://en.wikipedia.org/wiki/Matrix_(mathematics)

http://en.wikipedia.org/wiki/Algorithm

http://en.wikipedia.org/wiki/Gaussian_elimination

http://en.wikipedia.org/wiki/Quicksort

http://en.wikipedia.org/wiki/Simplex_algorithm


A=[10.^-16 59.17;5.29 -6.13];b=[59.17 46.78]';B=[A b];

fGauss.m

backward.m

B

0.0 1.0000

ans =

10.0019 1.0000

B=fGauss(B)

x=backward(B)


B =

10.^-16 59.1400 59.1700 5.2900 -6.1300 46.7800

FindExchange row 1 and row 2

2900.5},max{ 2111 bb

temp=B(1,:);B(1,:)=B(2,:);B(2,:)=temp;


U=fGauss(B);x=backward(U)

x =

10.0019 1.0000

B =

5.2900 -6.1300 46.7800 10.^-16 59.1400 59.1700


78.4613.6291.5

59170059140030

21

21

xx

xx

A=[30.00 591400;5.29 -6.13];b=[591700 46.78]';B=[A b];


Scaled partial pivotingScaled partial pivoting

kjj

k as max

k

ki

ik s

a

maxargp where

pivotith theas pselect

Exchange row i and row p


A =

2.1100 -4.2100 0.9210 4.0100 10.2000 -1.1200 1.0900 0.9870 0.8320

Determine the first pivot by[v p]=max(A(:,1)./s);

>> s=max(abs(A'))'

s =

4.2100 10.2000 1.0900

p =

3


Scaled partial pivotingScaled partial pivoting

Function [B,pr]=pss(A,b)Function [B,pr]=pss(A,b)n=length(b);B=[A b]; pr=[]; n=length(b);B=[A b]; pr=[]; s=max(abs(A'))' Pr=1:n;Pr=1:n;for i= 1:nfor i= 1:n

A.A. Find the ith pivot row and set it to pFind the ith pivot row and set it to pB.B. Exchange pr(i) and pr(p)Exchange pr(i) and pr(p)C.C. Exchange row i and row p of matrix BExchange row i and row p of matrix BD.D. For each row j > iFor each row j > i

1)1) Set d to the ratio B(j,i) / B(i,i)Set d to the ratio B(j,i) / B(i,i)2)2) Set B(j,:) to B(j,:)-d*B(i,:)Set B(j,:) to B(j,:)-d*B(i,:)

Return B and prReturn B and pr


ExampleExampleA=[2.11 -4.21 .921;4.01 10.2 -1.12;1.09 .987 .832];b=[2.01 -3.09 4.21]';

spp.m[B,pr]=spp(A,b)

B =

1.0900 0.9870 0.8320 4.2100 0 -6.1206 -0.6896 -6.1396 0 0.0000 -4.9209 -25.1675

pr =

3 1 2


ExampleExampleA=[1.19 2.11 -100 1;14.2 -0.122 12.2 -1;0 100 -99.9 1; 15.3 0.110 -13.1 -1];b=[1.12 3.44 2.15 4.16]';

[B,pr]=spp(A,b);x=backward(B);

x =

0.1768 0.0127 -0.0207 -1.1826


ExerciseExercise

Implement the scaled partial pivoting Implement the scaled partial pivoting method to improve the forward method to improve the forward Gauss elimination method.Gauss elimination method.

Verify your matlab codes by two Verify your matlab codes by two examplesexamples


Avoid row swappingAvoid row swapping

Row swapping results in time Row swapping results in time consumingconsuming

No row swappingNo row swapping Use a vector v to emulate row swappingUse a vector v to emulate row swapping Swap elements in v instead of swapping Swap elements in v instead of swapping

rows in Brows in B How to revise spp.m to avoid row How to revise spp.m to avoid row

swapping ? swapping ?

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Numerical method1 Iterative approaches for solving linear systems Successive over-relaxation Post-nonlinear system Partial pivoting Lecture 11-II