LINBCG

The LINBCG function is used in conjunction with SPRSIN to solve a set of n linear equations with n unknowns using the iterative biconjugate gradient method. The result is an n -element vector.

LINBCG is based on the routine


linbcg

described in section 2.7 of Numerical Recipes in C: The Art of Scientific Computing (Second Edition), published by Cambridge University Press, and is used by permission.

Note: Numerical Recipes recommends using double-precision arithmetic to perform this computation.

Arguments

A

A row-indexed sparse array created by the SPRSIN function.

B

An n -element vector containing the right-hand side of the linear system Ax=b .

X

An n -element vector containing the initial solution of the linear system.

Keywords

DOUBLE

Set this keyword to force the computation to be done in double-precision arithmetic.

ITOL

Use this keyword to specify which convergence test should be used. Set ITOL to one of the following:

Iteration stops when Ù A x - b Ù / Ù b Ù is less than the value specified by TOL.
Iteration stops when Ù Ã^-1 ( A x - b )Ù / Ù Ã^-1 b Ù (where Ã is a "preconditioning" matrix close to A ) is less than the value specified by TOL.
The routine uses its own estimate of error in x . Iteration stops when the magnitude of the error divided by the magnitude of x is less than the value specified by TOL. This is the default setting.
The same as 3, except that the routine uses the largest (in absolute value) component of the error and the largest component of x rather than the vector magnitudes.

TOL

Use this keyword to specify the desired convergence tolerance. For single-precision calculations, the default value is 1.0 ¥ 10^-7 . For double-precision values, the default is 1.0 ¥ 10^-14 .

ITER

Use this keyword to specify an output variable that will be set to the number of iterations performed.

ITMAX

The maximum allowed number of iterations. The default is n ² .

Example

Begin with an array A:

A = [[ 5.0, 0.0, 0.0, 1.0, -2.0], $

[ 3.0, -2.0, 0.0, 1.0, 0.0], $

[ 4.0, -1.0, 0.0, 2.0, 0.0], $

[ 0.0, 3.0, 3.0, 1.0, 0.0], $

[-2.0, 0.0, 0.0, -1.0, 2.0]]

B = [7.0, 1.0, 3.0, 3.0, -4.0] ; Define a right-hand side vector B.

X = REPLICATE(1.0, N_ELEMENTS(B)) ; Start with an initial guess at the solution.

result = LINBCG(SPRSIN(A), B, X)
; Solve the linear system Ax=b.

PRINT, result ; Print the result.

IDL prints:

1.00000 1.00000 -5.53459e-08 3.02525e-07 -1.00000

The exact solution is [1, 1, 0, 0, -1].