Calculate the L1 Norm of a Square Matrix

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

Routine Name: mat_1norm

Author: Christian Bolander

Language: Fortran. This code can be compiled using the GNU Fortran compiler by $ gfortran -c mat_1norm.f90

and can be added to a program using

$ gfortran program.f90 mat_1norm.o l1_vec_norm.o

Description/Purpose: This routine calculates the -norm of an arbitrary square matrix, A. This is done by finding the maximum column -norm, i.e.

Input:

n : INTEGER - the number of rows and columns in the square matrix, A

A : REAL - a square matrix of size n x n

Output:

norm : REAL - the -norm of the matrix, A.

Usage/Example:

This routine can be implemented in a program as follows

n = 3
ALLOCATE(A(1:n, 1:n), Approx(1:n, 1:n))
A(1, 1) = 1D0
A(1, 2) = 2D0
A(1, 3) = 3D0
A(2, 1) = 4D0
A(2, 2) = 5D0
A(2, 3) = 6D0
A(3, 1) = 7D0
A(3, 2) = 8D0
A(3, 3) = 9D0
Approx(1, 1) = 0.44D0
Approx(1, 2) = 2.36D0
Approx(1, 3) = 3.04D0
Approx(2, 1) = 3.09D0
Approx(2, 2) = 5.87D0
Approx(2, 3) = 6.66D0
Approx(3, 1) = 7.36D0
Approx(3, 2) = 7.77D0
Approx(3, 3) = 9.07D0
CALL mat_1norm(A - Approx, n, norm)
WRITE(*,*) norm

The output from the above code:

   1.8300000000000005       

which is the -norm of the matrix A.

Implementation/Code: The code for mat_1norm can be seen below. Note that the l1_vec_norm subroutine is also called

SUBROUTINE mat_1norm(A, n, norm)
	IMPLICIT NONE
	
	! I/O variables
	INTEGER, INTENT(IN) :: n
	REAL*8, INTENT(IN) :: A(1:n, 1:n)
	REAL*8, INTENT(OUT) :: norm
	
	! Initialize subroutine variables
	INTEGER :: i
	REAL*8 :: col_1norm
	
	norm = 0.D0
	col_1norm = 0.D0
	
	! Loop through each column in the matrix A and find the l1 vector
	! norm for that column.
	DO i = 1, n
		CALL l1_vec_norm(A(:, i), n, col_1norm)
		! If a new maximum value for the column l1 norm has been found,
		! save it
		IF (col_1norm > norm) THEN
			norm = col_1norm
		ENDIF
	END DO
	
END SUBROUTINE