Solve a Square (Diagonal) Linear System

This is a part of the student software manual project for Math 5610: Computational Linear Algebra and Solution of Systems of Equations.

Routine Name: lss_diag

Author: Christian Bolander

Language: Fortran. This code can be compiled using the GNU Fortran compiler by

$ gfortran -c lss_diag.f90

and can be added to a program using

$ gfortran program.f90 lss_diag.o

Description/Purpose: This routine computes the solution of a square, linear system of equations

where the coefficient matrix, A is a diagonal matrix, i.e.

Input:

m : INTEGER - number of rows and columns in the matrix A, and the length of b and x

A : REAL - square, diagonal matrix of size m x m

b : REAL - arbitrary vector of length m

Output:

x : REAL - the solution vector of length m

Usage/Example:

This routine can be implemented in a program as follows

INTEGER :: m, i
REAL*8, ALLOCATABLE :: A(:, :)
REAL*8, ALLOCATABLE :: x(:), b(:)

m = 3
ALLOCATE(A(1:m, 1:m), b(1:m), x(1:m))
A = RESHAPE((/1.D0, 0.D0, 0.D0, &
			& 0.D0, 4.D0, 0.D0, &
			& 0.D0, 0.D0, 7.D0/), (/m, m/), ORDER=(/2, 1/))
b = (/1.D0, 2.D0, 3.D0/)
CALL lss_diag(m, A, b, x)
WRITE(*,*) x

The outputs from the above code:

   1.0000000000000000       0.50000000000000000       0.42857142857142855 

Implementation/Code: The code for lss_diag can be seen below.

SUBROUTINE lss_diag(m, A, b, x)
	IMPLICIT NONE
	
	! Takes as an input a matrix `A` of size `m` x `m` and a vector `b`
	! of length `m`. Out puts a solution vector `x` of length `m` that
	! solves the system of equations `A``x`=`b`. This is a special case
	! where `A` is a diagonal matrix.
	INTEGER, INTENT(IN) :: m
	REAL*8, INTENT(IN) :: A(1:m, 1:m), b(1:m)
	REAL*8, INTENT(OUT) :: x(1:m)
	
	! Initialize an increment variable.
	INTEGER :: i
	
	! Find the solution to the linear system of equations by simply
	! dividing the value in `b` by the corresponding diagonal element
	! in `A`.
	DO i = 1, m
		x(i) = b(i)/A(i, i)
	END DO
END SUBROUTINE