linsolve

Arguments

A

a na x ma real matrix (possibly sparse)

b

a na x 1 vector (same row dimension as A)

x0

a real vector

kerA

a ma x k real matrix

Description

linsolve computes all the solutions to A*x+b=0.

x0 is a particular solution (if any) and kerA=nullspace of A. Any x=x0+kerA*w with arbitrary w satisfies A*x+b=0.

If compatible x0 is given on entry, x0 is returned. If not a compatible x0, if any, is returned.

Examples

A=rand(5,3)*rand(3,8);
b=A*ones(8,1);[x,kerA]=linsolve(A,b);A*x+b   //compatible b
b=ones(5,1);[x,kerA]=linsolve(A,b);A*x+b   //uncompatible b
A=rand(5,5);[x,kerA]=linsolve(A,b), -inv(A)*b  //x is unique

// Benchmark with other linear sparse solver:
[A,descr,ref,mtype] = ReadHBSparse(SCI+"/modules/umfpack/demos/bcsstk24.rsa");

b = zeros(size(A,1),1);

tic();
res = umfpack(A,'\',b);
mprintf('\ntime needed to solve the system with umfpack: %.3f\n',toc());

tic();
res = linsolve(A,b);
mprintf('\ntime needed to solve the system with linsolve: %.3f\n',toc());

tic();
res = A\b;
mprintf('\ntime needed to solve the system with the backslash operator: %.3f\n',toc());

See also

• inv — matrix inverse
• pinv — pseudoinverse
• colcomp — column compression, kernel, nullspace
• im_inv — inverse image
• umfpack — solve sparse linear system
• backslash — (\) left matrix division: Exact or least square solution