GS_ITER

The GS_ITER function solves an n by n linear system of equations using Gauss-Seidel iteration with over- and under-relaxation to enhance convergence.

Note
The equations must be entered in diagonally dominant form to guarantee convergence. A system is diagonally dominant if the diagonal element in a given row is greater than the sum of the absolute values of the non-diagonal elements in that row.

This routine is written in the IDL language. Its source code can be found in the file gs_iter.pro in the lib subdirectory of the IDL distribution.

Syntax

Result = GS_ITER( A, B [, /CHECK] [, /DOUBLE] [, LAMBDA=value{0.0 to 2.0}] [, MAX_ITER=value] [, TOL=value] [, X_0=vector] )

Return Value

Returns the solution to the linear system of equations of the specified dimensions.

Arguments

A

An n by n integer, single-, or double-precision floating-point array. On output, A is divided by its diagonal elements. Integer input values are converted to single-precision floating-point values.

B

A vector containing the right-hand side of the linear system Ax=b. On output, B is divided by the diagonal elements of A.

Keywords

CHECK

Set this keyword to check the array A for diagonal dominance. If A is not in diagonally dominant form, GS_ITER reports the fact but continues processing on the chance that the algorithm may converge.

DOUBLE

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

LAMBDA

A scalar value in the range: [0.0, 2.0]. This value determines the amount of relaxation. Relaxation is a weighting technique used to enhance convergence.

If LAMBDA = 1.0, no weighting is used. This is the default.

If 0.0 ≤ LAMBDA < 1.0, convergence improves in oscillatory and non-convergent systems.

If 1.0 < LAMBDA ≤ 2.0, convergence improves in systems already known to converge.

MAX_ITER

The maximum allowed number of iterations. The default value is 30.

TOL

The relative error tolerance between current and past iterates calculated as: ½( (current-past)/current )½. The default is 1.0 × 10^-4.

X_0

An n-element vector that provides the algorithm's starting point. The default is [1.0, 1.0, ... , 1.0].

Example

; Define an array A:
A = [[ 1.0, 7.0, -4.0], $
[ 4.0, -4.0, 9.0], $
[12.0, -1.0, 3.0]]

; Define the right-hand side vector B:
B = [12.0, 2.0, -9.0]

; Compute the solution to the system:
RESULT = GS_ITER(A, B, /CHECK)

IDL prints:

Input matrix is not in Diagonally Dominant form. 
Algorithm may not converge. 
% GS_ITER: Algorithm failed to converge within given parameters.

Since the A represents a system of linear equations, we can reorder it into diagonally dominant form by rearranging the rows:

A = [[12.0, -1.0, 3.0], $
[ 1.0, 7.0, -4.0], $
[ 4.0, -4.0, 9.0]]

; Make corresponding changes in the ordering of B:
B = [-9.0, 12.0, 2.0]

; Compute the solution to the system:
RESULT = GS_ITER(A, B, /CHECK)
print, RESULT