I am not very good with MATLAB. Here is the problem statement & my attempt at a

Question

I am not very good with MATLAB. Here is the problem statement & my attempt at a solution, please find the errors in the code so that it will perform the operation prescribed. Please do not write your own code from scratch, adjust the one given accordingly. [One of the problems I noticed was that the matrix I created was not filled with actual numbers, just the words 'double']

The code must include a working 10x10 example. (answer with output included will be favored)

function y = btridiag( a, b, c, f )

% Solves the [nxn] block - tridiagonal system for y

A = cell(5,5); %%create [5x5] matrix A

for i=1:5

for j=1:5

A{i,j}=rand(2,2); %%fill matrix A with random [2x2] submatrices

end

end

B = cell(5,1); %%create [5x1] matrix B

for i=1:5

for j=1:1

B{i,j}=rand(2,1); %%fill matrix B with random [2x1] submatrices

end

end

n=5;

a = rand(2,2);

b = rand(2,2);

c = rand(2,2);

f = rand(2,2);%% a,b,c,f are randomly generated vectors of length n

n = length(f);

B = zeros(n,1);

y(1) = B(1); %%define variables

w(1) = 1/a(1);

for i=2:n %recursive algorithm for value of x(i)

y(i) = B(i)-c(i)*w(i-1)*y(i-1);

w(i) = (a(i)-c(i)*w(i-1)*b(i-1))^-1;

end %%End forward sweep

x(n) = w(n)*y(n); %%Compute value of x3

for j=1:n-1;

k = n-j;

x(k) = w(k)*y(k)-w(k)*b(k)*x(k+1);

end %%End backward sweep, plug back in

%%Example

diagA(i, j) = a(i,j);

diagA(i, j+1) = b(i,j);

diagA(i-1, j) = c(i,j);

y = btridiag( a, b, c, f );

Write a computer program to solve a banded matrix, using the block-tridiagonal matrix. It should be general for an n dimensional system. It should have comments in the code explaining the stages of the algorithm. Use two (10 x 10) examples (one for each) to show their performance.

Explanation / Answer

Hello there I am attaching my code please go through it you will get the desired results

function A = blktridiag(Amd,Asub,Asup,n)

% BLKTRIDIAG: computes a sparse (block) tridiagonal matrix with n blocks

% usage: A = BLKTRIDIAG(Amd,Asub,Asup,n) % identical blocks

% usage: A = BLKTRIDIAG(Amd,Asub,Asup) % a list of distinct blocks

%

% BLKTRIDIAG runs in two distinct modes. The first mode

% supplies three blocks, one for the main diagonal, and

% the super and subdiagonal blocks. These blocks will be

% replicated n times down the main diagonals of the matrix,

% and n-1 times down the sub and super diagonals.

%

% The second mode is to supply a list of distinct blocks

% for each diagonal, as planes of 3d arrays. No replication

% factor is needed in this mode.

%

% arguments: (input mode 1)

% Amd - pxq array, forming the main diagonal blocks

%

% Asub - pxq array, sub diagonal block

% Asub must be the same size and shape as Amd

%

% Asup - pxq array, super diagonal block

% Asup must be the same size and shape as Amd

%

% n - scalar integer, defines the number of blocks

% When n == 1, only a single block will be formed, A == Amd

%

% arguments: (input mode 2)

% Amd - pxqxn array, a list of n distinct pxq arrays

% Each plane of Amd corresponds to a single block

% on the main diagonal.

%

% Asub - pxqx(n-1) array, a list of n-1 distinct pxq arrays,

% Each plane of Asub corresponds to a single block

% on the sub-diagonal.

%

% Asup - pxqx(n-1) array, a list of n-1 distinct pxq arrays,

% Each plane of Asup corresponds to a single block

% on the super-diagonal.

%

% Note: the sizes of Amd, Asub, and Asup must be consistent

% with each other, or an error will be generated.

%

% arguments: (output)

% A - (n*p by n*q) SPARSE block tridiagonal array

% If you prefer that A be full, use A=full(A) afterwards.

%

%

% Example 1:

% Compute the simple 10x10 tridiagonal matrix, with 2 on the

% diagonal, -1 on the off diagonal.

%

% A = blktridiag(2,-1,-1,10);

%

%

% Example 2:

% Compute the 5x5 lower bi-diagonal matrix, with blocks of

% [1 1;1 1] on the main diagonal, [2 2;2 2] on the sub-diagonal,

% and blocks of zeros above.

%

% A = blktridiag(ones(2),2*ones(2),zeros(2),5);

%

% Example 3:

% Compute the 3x6 tridiagonal matrix, with non-square blocks

% that vary along the main diagonal, [2 2] on the sub-diagonal,

% and [1 1] on the super-diagonal. Note that all blocks must have

% the same shape.

%

% A = blktridiag(rand(1,2,3),2*ones(1,2,2),ones(1,2,2));

%

%

% Which mode of operation are we in?

if nargin==4

% replicated block mode

  

% verify the inputs in this mode are 2-d arrays.

if (length(size(Amd))~=2) || ...

(length(size(Asub))~=2) || ...

(length(size(Asup))~=2)

error 'Inputs must be 2d arrays if a replication factor is provided'

end

  

% get block sizes, check for consistency

[p,q] = size(Amd);

if isempty(Amd)

error 'Blocks must be non-empty arrays or scalars'

end

if any(size(Amd)~=size(Asub)) || any(size(Amd)~=size(Asup))

error 'Amd, Asub, Asup are not identical in size'

end

if isempty(n) || (length(n)>1) || (n<1) || (n~=floor(n))

error 'n must be a positive scalar integer'

end

  

% scalar inputs?

% since p and q are integers...

if (p*q)==1

if n==1

A = Amd;

else

% faster as Jos points out

A = spdiags(repmat([Asub Amd Asup],n,1),-1:1,n,n);

end

% no need to go any farther

return

end

  

% use sparse. the main diagonal elements of each array are...

v = repmat(Amd(:),n,1);

% then the sub and super diagonal blocks.

if n>1

% sub-diagonal

v=[v;repmat(Asub(:),n-1,1)];

  

% super-diagonal

v=[v;repmat(Asup(:),n-1,1)];

end

  

elseif nargin==3

% non-replicated blocks, supplied as planes of a 3-d array

  

% get block sizes, check for consistency

[p,q,n] = size(Amd);

if isempty(Amd)

error 'Blocks must be (non-empty) arrays or scalars'

end

  

if (p~=size(Asub,1)) || (q~=size(Asub,2)) || (p~=size(Asup,1)) || (q~=size(Asup,2))

error 'Amd, Asub, Asup do not have the same size blocks'

end

if (n>1) && (((n-1) ~= size(Asub,3)) || ((n-1) ~= size(Asup,3)))

error 'Asub and Asup must each have one less block than Amd'

end

  

% scalar inputs?

if (p*q)==1

if n==1

A = Amd(1);

else

% best to just use spdiags

A = spdiags([[Asub(:);0], Amd(:), [0;Asup(:)]],-1:1,n,n);

end

% no need to go any farther

return

end

  

% The main diagonal elements

v = Amd(:);

% then the sub and super diagonal blocks.

if n>1

% sub-diagonal

v=[v;Asub(:)];

% super-diagonal

v=[v;Asup(:)];

end

else

% must have 3 or 4 arguments

error 'Must have either 3 or 4 arguments to BLKTRIDIAG'

end

% now generate the index arrays. first the main diagonal

[ind1,ind2,ind3]=ndgrid(0:p-1,0:q-1,0:n-1);

rind = 1+ind1(:)+p*ind3(:);

cind = 1+ind2(:)+q*ind3(:);

% then the sub and super diagonal blocks.

if n>1

% sub-diagonal

[ind1,ind2,ind3]=ndgrid(0:p-1,0:q-1,0:n-2);

rind = [rind;1+p+ind1(:)+p*ind3(:)];

cind = [cind;1+ind2(:)+q*ind3(:)];

% super-diagonal

rind = [rind;1+ind1(:)+p*ind3(:)];

cind = [cind;1+q+ind2(:)+q*ind3(:)];

end

% build the final array all in one call to sparse

A = sparse(rind,cind,v,n*p,n*q);

% ====================================

% end mainline

% ====================================

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