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.

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

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.

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

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)

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)

-0.999982      2.99988      1.99994