NORM

The NORM function computes the norm of a vector or a two-dimensional array.

By default, NORM computes the L₂ (Euclidean) norm for vectors, and the L_∞ norm for arrays. You may use the LNORM keyword to specify different norms.

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

Syntax

Result = NORM( A [, /DOUBLE] [, LNORM={0 | 1 | 2 | n}])

Return Value

Returns the Euclidean or infinity norm of a vector or an array. This function always returns a float or double value.

Arguments

A

A can be either a real or complex vector, or a real or complex two-dimensional array.

Keywords

DOUBLE

Set this keyword to force the result to be returned as double precision. The default is to return a single-precision result if the input is single precision, or double precision otherwise. Internally, all computations are done using double-precision arithmetic.

LNORM

Set this keyword to indicate which norm to compute. If A is a vector, then the possible values of this keyword are:

Table 16-1: LNORM Keyword Values (Vector)
Value	Description
0	Compute the norm, defined as MAX(ABS(A)).
1	Compute the L₁ norm, defined as TOTAL(ABS(A)).
2	Compute the L₂ norm, defined as SQRT(TOTAL(ABS(A)^2)). This is the default.
n	Compute the L_n norm, defined as (TOTAL(ABS(A)^n))^(1/n) where n is any number, float-point or integer.

If A is a two-dimensional array, then the possible values of this keyword are:

Table 16-2: LNORM Keyword Values (Two-Dimensional Array)
Value	Description
0	Compute the L_∞ norm (the maximum absolute row sum norm), defined as MAX(TOTAL(ABS(A), 1)). This is the default.
1	Compute the L₁ norm (the maximum absolute column sum norm), defined as MAX(TOTAL(ABS(A), 2)).
2	Compute the L₂ norm (the spectral norm) defined as the largest singular value, computed from LA_SVD.

Examples

; Define an n-element complex vector A:
A = [COMPLEX(1, 0), COMPLEX(2,-2), COMPLEX(-3,1)]

; Compute the Euclidean norm of A and print:
PRINT, 'Euclidian Norm of A =', NORM(A)

; Define an m by n complex array B:
B = [[COMPLEX(1, 0), COMPLEX(2,-2), COMPLEX(-3,1)], $
[COMPLEX(1,-2), COMPLEX(2, 2), COMPLEX(1, 0)]]

;Compute the Infinity norm of B and print.
PRINT, 'Infinity Norm of B =', NORM(B, /DOUBLE)

IDL prints:

Euclidian Norm of A =    4.35890 
Infinity Norm of B =    6.9907048

Version History


Pre-4.0	Introduced