MATLAB Function Reference  lsqr

LSQR method

Syntax

• ```x = lsqr(A,b)
lsqr(A,b,tol)
lsqr(A,b,tol,maxit)
lsqr(A,b,tol,maxit,M)
lsqr(A,b,tol,maxit,M1,M2)
lsqr(A,b,tol,maxit,M1,M2,x0)
[x,flag] = lsqr(A,b,...)
[x,flag,relres] = lsqr(A,b,...)
[x,flag,relres,iter] = lsqr(A,b,...)
[x,flag,relres,iter,resvec] = lsqr(A,b,...)
[x,flag,relres,iter,resvec,lsvec] = lsqr(A,b,...)
```

Description

```x = lsqr(A,b) ``` attempts to solve the system of linear equations `A*x=b` for `x` if `A` is consistent, otherwise it attempts to solve the least squares solution `x` that minimizes `norm(b-A*x)`. The `m`-by-`n` coefficient matrix `A` need not be square but it should be large and sparse. The column vector `b` must have length `m`. `A` can be a function handle `afun` such that `afun(x,'notransp')` returns `A*x` and `afun(x,'transp')` returns `A'*x`. See Function Handles in the MATLAB Programming documentation for more information.

Parameterizing Functions Called by Function Functions, in the MATLAB Mathematics documentation, explains how to provide additional parameters to the function `afun`, as well as the preconditioner function `mfun` described below, if necessary.

If `lsqr` converges, a message to that effect is displayed. If `lsqr` fails to converge after the maximum number of iterations or halts for any reason, a warning message is printed displaying the relative residual `norm(b-A*x)/norm(b)` and the iteration number at which the method stopped or failed.

```lsqr(A,b,tol) ``` specifies the tolerance of the method. If `tol` is `[]`, then `lsqr` uses the default, `1e-6`.

```lsqr(A,b,tol,maxit) ``` specifies the maximum number of iterations. If `maxit` is `[]`, then `lsqr` uses the default, `min([m,n,20])`.

```lsqr(A,b,tol,maxit,M1) and lsqr(A,b,tol,maxit,M1,M2) ``` use `n`-by-`n` preconditioner `M` or `M = M1*M2` and effectively solve the system `A*inv(M)*y = b` for `y`, where `x = M*y`. If `M` is `[]` then `lsqr` applies no preconditioner. `M` can be a function `mfun` such that `mfun(x,'notransp')` returns `M\x` and `mfun(x,'transp')` returns `M'\x`.

```lsqr(A,b,tol,maxit,M1,M2,x0) ``` specifies the `n`-by-`1` initial guess. If `x0` is `[]`, then `lsqr` uses the default, an all zero vector.

```[x,flag] = lsqr(A,b,tol,maxit,M1,M2,x0) ``` also returns a convergence flag.

 Flag Convergence `0` `lsqr `converged to the desired tolerance `tol` within `maxit `iterations. `1` `lsqr` iterated `maxit` times but did not converge. `2` Preconditioner `M` was ill-conditioned. `3` `lsqr` stagnated. (Two consecutive iterates were the same.) `4` One of the scalar quantities calculated during `lsqr `became too small or too large to continue computing.

Whenever `flag` is not `0`, the solution `x` returned is that with minimal norm residual computed over all the iterations. No messages are displayed if you specify the `flag` output.

```[x,flag,relres] = lsqr(A,b,tol,maxit,M1,M2,x0) ``` also returns an estimate of the relative residual `norm(b-A*x)/norm(b)`. If `flag` is `0`, `relres <= tol`.

```[x,flag,relres,iter] = lsqr(A,b,tol,maxit,M1,M2,x0) ``` also returns the iteration number at which `x` was computed, where `0 <= iter <= maxit`.

```[x,flag,relres,iter,resvec] = lsqr(A,b,tol,maxit,M1,M2,x0) ``` also returns a vector of the residual norm estimates at each iteration, including `norm(b-A*x0)`.

```[x,flag,relres,iter,resvec,lsvec] = lsqr(A,b,tol,maxit,M1,M2,x0) ``` also returns a vector of estimates of the scaled normal equations residual at each iteration: `norm((A*inv(M))'*(B-A*X))/norm(A*inv(M),'fro')`. Note that the estimate of `norm(A*inv(M),'fro')` changes, and hopefully improves, at each iteration.

Examples

Example 1.

• ```n = 100;
on = ones(n,1);
A = spdiags([-2*on 4*on -on],-1:1,n,n);
b = sum(A,2);
tol = 1e-8;
maxit = 15;
M1 = spdiags([on/(-2) on],-1:0,n,n);
M2 = spdiags([4*on -on],0:1,n,n);

x = lsqr(A,b,tol,maxit,M1,M2);
```

displays the following message:

• ```lsqr converged at iteration 11 to a solution with relative
residual 3.5e-009
```

Example 2.

This example replaces the matrix `A` in Example 1 with a handle to a matrix-vector product function `afun`. The example is contained in an M-file `run_lsqr` that

• Calls `lsqr` with the function handle `@afun` as its first argument.
• Contains `afun` as a nested function, so that all variables in `run_lsqr` are available to `afun`.

The following shows the code for `run_lsqr`:

• ```function x1 = run_lsqr
n = 100;
on = ones(n,1);
A = spdiags([-2*on 4*on -on],-1:1,n,n);
b = sum(A,2);
tol = 1e-8;
maxit = 15;
M1 = spdiags([on/(-2) on],-1:0,n,n);
M2 = spdiags([4*on -on],0:1,n,n);
x1 = lsqr(@afun,b,tol,maxit,M1,M2);

function y = afun(x,transp_flag)
if strcmp(transp_flag,'transp')      % y = A'*x
y = 4 * x;
y(1:n-1) = y(1:n-1) - 2 * x(2:n);
y(2:n) = y(2:n) - x(1:n-1);
elseif strcmp(transp_flag,'notransp') % y = A*x
y = 4 * x;
y(2:n) = y(2:n) - 2 * x(1:n-1);
y(1:n-1) = y(1:n-1) - x(2:n);
end
end
end
```

When you enter

• ```x1=run_lsqr;
```

MATLAB displays the message

• ```lsqr converged at iteration 11 to a solution with relative
residual 3.5e-009
```

`bicg`, `bicgstab`, `cgs`, `gmres`, `minres`, `norm`, `pcg`, `qmr`, `symmlq`, `function_handle` (`@`)