How to solve Ax=b when some values of x are known and b is vector of zeros - matlab

Question

I have a matrix Q=A (64x64), a vector f=b is vector of zeros and I know some values of x=q. I know that I should transfer the corresponding column and row (of known x=q) on the right side of the equation (to b), but I don't know, how to make it in Matlab. I should to do that for 1st, 5th, 9th, 13th, 17th, 21st, 25th, 29th, 33th, 37th, 41st, 45th, 49th, 53th, 57th and 61st, column and row. Can you help me, pls?
This is the program:
clear all;
K=zeros(64,64);
f=zeros(64,1);
ne=32;
E= 8000; %Young
P= 0.51; %Poisson
Lambda=(E*P)/((1+P)*(1-2*P));
Eta=E/(2*(1+P));
%ILOK
ILOK=[
1 3 5 7 2 4 6 8;
5 7 9 11 6 8 10 12;
9 11 13 15 10 12 14 16;
13 15 17 19 14 16 18 20;
17 19 21 23 18 20 22 24;
21 23 25 27 22 24 26 28;
25 27 29 31 26 28 30 32;
29 31 33 35 30 32 34 36;
33 35 37 39 34 36 38 40;
37 39 41 43 38 40 42 44;
41 43 45 47 42 44 46 48;
45 47 49 51 46 48 50 52;
49 51 53 55 50 52 54 56;
53 55 57 59 54 56 58 60;
57 59 61 63 58 60 62 64;
61 63 1 3 62 64 2 4;
3 0 7 0 4 0 8 0;
7 0 11 0 8 0 12 0;
11 0 15 0 12 0 16 0;
15 0 19 0 16 0 20 0;
19 0 23 0 20 0 24 0;
23 0 27 0 24 0 28 0;
27 0 31 0 28 0 32 0;
31 0 35 0 32 0 36 0;
35 0 39 0 36 0 40 0;
39 0 43 0 40 0 44 0;
43 0 47 0 44 0 48 0;
47 0 51 0 48 0 52 0;
51 0 55 0 52 0 56 0;
55 0 59 0 56 0 60 0;
59 0 63 0 60 0 64 0;
63 0 3 0 64 0 4 0;
];
%x
xm=[
9.000 14.500 8.315 13.396;
8.315 13.396 6.364 10.253;
6.364 10.253 3.444 5.549;
3.444 5.549 0.000 0.000;
0.000 0.000 -3.444 -5.549;
-3.444 -5.549 -6.364 -10.253;
-6.364 -10.253 -8.315 -13.396;
-8.315 -13.396 -9.000 -14.500;
-9.000 -14.500 -8.315 -13.396;
-8.315 -13.396 -6.364 -10.253;
-6.364 -10.253 -3.444 -5.549;
-3.444 -5.549 0.000 0.000;
0.000 0.000 3.444 5.549;
3.444 5.549 6.364 10.253;
6.364 10.253 8.315 13.396;
8.315 13.396 9.000 14.500;
14.500 20.000 13.396 18.748;
13.396 18.748 10.253 14.142;
10.253 14.142 5.549 7.654;
5.549 7.654 0.000 0.000;
0.000 0.000 -5.549 -7.654;
-5.549 -7.654 -10.253 -14.142;
-10.253 -14.142 -13.396 -18.748;
-13.396 -18.748 -14.500 -20.000;
-14.500 -20.000 -13.396 -18.748;
-13.396 -18.748 -10.253 -14.142;
-10.253 -14.142 -5.549 -7.654;
-5.549 -7.654 0.000 0.000;
0.000 0.000 5.549 7.654;
5.549 7.654 10.253 14.142;
10.253 14.142 13.396 18.748;
13.396 18.748 14.500 20.000;
];
%y
ym=[
0.000 0.000 3.444 5.549
3.444 5.549 6.364 10.253
6.364 10.253 8.315 13.396
8.315 13.396 9.000 14.500
9.000 14.500 8.315 13.396
8.315 13.396 6.364 10.253
6.364 10.253 3.444 5.549
3.444 5.549 0.000 0.000
0.000 0.000 -3.444 -5.549
-3.444 -5.549 -6.364 -10.253
-6.364 -10.253 -8.315 -13.396
-8.315 -13.396 -9.000 -14.500
-9.000 -14.500 -8.315 -13.396
-8.315 -13.396 -6.364 -10.253
-6.364 -10.253 -3.444 -5.549
-3.444 -5.549 0.000 0.000
0.000 0.000 5.549 7.654
5.549 7.654 10.253 14.142
10.253 14.142 13.396 18.748
13.396 18.748 14.500 20.000
14.500 20.000 13.396 18.748
13.396 18.748 10.253 14.142
10.253 14.142 5.549 7.654
5.549 7.654 0.000 0.000
0.000 0.000 -5.549 -7.654
-5.549 -7.654 -10.253 -14.142
-10.253 -14.142 -13.396 -18.748
-13.396 -18.748 -14.500 -20.000
-14.500 -20.000 -13.396 -18.748
-13.396 -18.748 -10.253 -14.142
-10.253 -14.142 -5.549 -7.654
-5.549 -7.654 0.000 0.000
];
%Ke a fe of element
for k=1:ne
x=xm(k,:);%k-ty radek x-ove matice
y=ym(k,:);%k-ty radek y-ove matice
Au=zeros(4,4);
Av=zeros(4,4);
Auv=zeros(4,4);
Avu=zeros(4,4);
%Numerical integration
for i=1:9
a=0.774596669241483;
gaus=[1 0 0 68/81;
2 0 a 40/81;
3 a 0 40/81;
4 0 -a 40/81;
5 -a 0 40/81;
6 a a 25/81;
7 a -a 25/81;
8 -a -a 25/81;
9 -a a 25/81];
r=gaus(i,2);
s=gaus(i,3);
N=[(1/4)*(1-r)*(1-s);
(1/4)*(1+r)*(1-s);
(1/4)*(1+r)*(1+s);
(1/4)*(1-r)*(1+s)];
Nr=[(1/4)*(s-1);
(1/4)*(1-s);
(1/4)*(s+1);
(1/4)*(-s-1)];
Ns=[(1/4)*(r-1);
(1/4)*(-1-r);
(1/4)*(r+1);
(1/4)*(1-r)];
%Jacob matrix
j1=Nr'*x';
j2=Nr'*y';
j3=Ns'*x';
j4=Ns'*y';
J=[j1 j2;
j3 j4];
detJ=abs(det(J));
invJ=inv(J);
%Nx a Ny
Nx=invJ(1,1)*Nr+invJ(1,2)*Ns;
Ny=invJ(2,1)*Nr+invJ(2,2)*Ns;
ds=gaus(i,4)*detJ;
Au=Au+(Nx*(Lambda*Nx'+2*Eta*Nx')+Eta*Ny*Ny')*ds;
Av=Av+(Ny*(Lambda*Ny'+2*Eta*Ny')+Eta*Nx*Nx')*ds;
Auv=Auv+(Nx*Lambda*Ny'+Eta*Ny*Nx')*ds;
Avu=Avu+(Ny*Lambda*Nx'+Eta*Nx*Ny')*ds;
Ke=[Au Auv;
Avu Av];
fe=zeros(8,1);
end
%K a f
N=8;
je=1:N;
mg(je)=ILOK(k,je);
igl=mg;
inen=find(igl);
K(igl(inen),igl(inen))=K(igl(inen),igl(inen))+Ke(igl>0,igl>0);
f(igl(inen))=f(igl(inen))+fe(igl>0);
Ke=zeros(8,8);
fe=zeros(8,1);
end
K;
And then I need to solve q=K/f
I would like transfer the columns a rows from this matrix to f (in this case).
Thank you for your help :)

Answer 1

This is known as solving for the null space of a matrix.
Z = null(A)
Z = null(A,'r')
Z = null(A) is an orthonormal basis for the null space of A obtained
from the singular value decomposition. That is, A*Z has negligible
elements, size(Z,2) is the nullity of A, and Z'*Z = I.
Z = null(A,'r') is a "rational" basis for the null space obtained from
the reduced row echelon form. A*Z is zero, size(Z,2) is an estimate
for the nullity of A, and, if A is a small matrix with integer
elements, the elements of the reduced row echelon form (as computed
using rref) are ratios of small integers.
The orthonormal basis is preferable numerically, while the rational
basis may be preferable pedagogically.
Please refer to the fully worked out examples in the References section below, as they include MATLAB-specific examples, and worked out "by hand" solutions.
Good luck!
References
http://www.mathworks.com/help/matlab/ref/null.html
http://www.math.sunysb.edu/~badger/mat211f12/solver.pdf