lsqr_module Module

User function to access the sparse matrix A.

main solver routine

number of rows in A matrix

number of columns in A matrix

number of nonzero elements in A matrix

sparsity row indices

sparsity column indices

sparse A matrix

relative error in definition of A

relative error in definition of b

An upper limit on cond(Abar), the apparent condition number of the matrix Abar.

max iterations

output unit for printing

A*x (dimension m)

A(transpose)*y (dimension n)

workspace array (dimension n)

workspace array (dimension n)

main solver routine

Constructor. Must be call first.

internal routine

number of rows in A matrix

number of columns in A matrix

nonzero elements of A

row indices of nonzero elements of A

column indices of nonzero elements of A

relative error in definition of A

relative error in definition of b

An upper limit on cond(Abar), the apparent condition number of the matrix Abar.

max iterations

output unit for printing

number of rows in A matrix

number of columns in A matrix

[n]

[m]

the computed solution x.

exit code (see LSQR).

the number of rows in A.

the number of columns in A.

The damping parameter for problem 3 above. (damp should be 0.0 for problems 1 and 2.) If the system A*x = b is incompatible, values of damp in the range 0 to sqrt(relpr)*norm(A) will probably have a negligible effect. Larger values of damp will tend to decrease the norm of x and reduce the number of iterations required by LSQR.

A logical variable to say if the array se(*) of standard error estimates should be computed. If m > n or damp > 0, the system is overdetermined and the standard errors may be useful. (See the first LSQR reference.) Otherwise (m <= n and damp = 0) they do not mean much. Some time and storage can be saved by setting wantse = .false. and using any convenient array for se(*), which won't be touched.

The rhs vector b. Beware that u is over-written by LSQR.

workspace

workspace

Returns the computed solution x.

If wantse is true, the dimension of se must be n or more. se(*) then returns standard error estimates for the components of x. For each i, se(i) is set to the value rnorm * sqrt( sigma(i,i) / t ), where sigma(i,i) is an estimate of the i-th diagonal of the inverse of Abar(transpose)*Abar and: * t = 1 if m <= n * t = m - n if m > n and damp = 0 * t = m if damp /= 0

An estimate of the relative error in the data defining the matrix A. For example, if A is accurate to about 6 digits, set atol = 1.0e-6.

An estimate of the relative error in the data defining the rhs vector b. For example, if b is accurate to about 6 digits, set btol = 1.0e-6.

An upper limit on cond(Abar), the apparent condition number of the matrix Abar. Iterations will be terminated if a computed estimate of cond(Abar) exceeds conlim. This is intended to prevent certain small or zero singular values of A or Abar from coming into effect and causing unwanted growth in the computed solution.

An upper limit on the number of iterations. Suggested value: * itnlim = n/2 for well-conditioned systems with clustered singular values, * itnlim = 4*n otherwise.

File number for printed output. If nonzero, a summary will be printed on file nout.

An integer giving the reason for termination:

The number of iterations performed.

An estimate of the Frobenius norm of Abar. This is the square-root of the sum of squares of the elements of Abar. If damp is small and if the columns of A have all been scaled to have length 1.0, anorm should increase to roughly sqrt(n). A radically different value for anorm may indicate an error in subroutine aprod (there may be an inconsistency between modes 1 and 2).

An estimate of cond(Abar), the condition number of Abar. A very high value of acond may again indicate an error in aprod.

An estimate of the final value of norm(rbar), the function being minimized (see notation above). This will be small if A*x = b has a solution.

An estimate of the final value of norm( Abar(transpose)*rbar ), the norm of the residual for the usual normal equations. This should be small in all cases. (arnorm will often be smaller than the true value computed from the output vector x.)

An estimate of the norm of the final solution vector x.

No. of rows of A.

No. of columns of A.

A file number for printed output.

The machine precision.

Error indicator. inform = 0 if aprod seems to be consistent. inform = 1 otherwise.

The number of rows in A.

The number of columns in A.

A file number for printed output. If nout = 0, nothing is printed.

An estimate of norm(A) or norm( A, delta*I ) if delta > 0. Normally this will be available from LSQR or CRAIG.

Possibly defines a damped problem.

Machine precision.

The right-hand side of Ax = b etc.

On exit, u = r (where r = b - Ax).

On exit, v = A'r.

On exit, w = A'r - damp^2 x.

The given estimate of a solution.

inform = 0 if b = 0 and x = 0. inform = 1, 2 or 3 if x seems to solve systems 1 2 or 3 above.

These are dimensionless quantities that should be "small" if x does seem to solve one of the systems. "small" means less than tol = eps**power, where power is defined as a parameter below.

These are dimensionless quantities that should be "small" if x does seem to solve one of the systems. "small" means less than tol = eps**power, where power is defined as a parameter below.

These are dimensionless quantities that should be "small" if x does seem to solve one of the systems. "small" means less than tol = eps**power, where power is defined as a parameter below.