The IMSL_MATRIX_NORM function computes various norms of a rectangular matrix, a matrix stored in band format, and a matrix stored in coordinate format.
This routine requires an IDL Advanced Math and Stats license. For more information, contact your sales or technical support representative.
By default, IMSL_MATRIX_NORM computes the Frobenius norm:
If the keyword One_Norm is used, the one norm
is returned. If the keyword Inf_Norm is used, the infinity norm
is returned.
Examples
Example 1
Compute the Frobenius norm, infinity norm, and one norm of matrix A.
a = TRANSPOSE([[1.0, 2.0, -2.0, 3.0], $
[-2.0, 1.0, 3.0, 0.0], [0.0, 3.0, 1.0, -7.0], $
[5.0, -2.0, 7.0, 6.0], [4.0, 3.0, 4.0, 0.0]])
frobenius_norm = IMSL_MATRIX_NORM(a)
inf_norm = IMSL_MATRIX_NORM(a, /INF_NORM)
one_norm = IMSL_MATRIX_NORM(a, /ONE_NORM)
PRINT, 'Frobenius norm = ', frobenius_norm
PRINT, 'Infinity norm = ', inf_norm
PRINT, 'One norm = ', one_norm
Frobenius norm = 15.6844
Infinity norm = 20.0000
One norm = 17.0000
Example 2
Compute the Frobenius norm, infinity norm, and one norm of matrix a. Matrix a is stored in band storage mode.
nlca = 1
nuca = 1
n = 4
a = [0.0, 2.0, 3.0, -1.0, 1.0, 1.0, 1.0, 1.0, 0.0, 3.0, 4.0, 0.0]
frobenius_norm = IMSL_MATRIX_NORM(n, nlca, nuca, a)
inf_norm = IMSL_MATRIX_NORM(n, nlca, nuca, a, /INF_NORM)
one_norm = IMSL_MATRIX_NORM(n, nlca, nuca, a, /ONE_NORM)
PRINT, 'Frobenius norm = ', frobenius_norm
PRINT, 'Infinity norm = ', inf_norm
PRINT, 'One norm = ', one_norm
Frobenius norm = 6.55744
Infinity norm = 5.00000
One norm = 8.00000
Example 3
Compute the Frobenius norm, infinity norm, and one norm of matrix a. Matrix a is stored in symmetric band storage mode.
nlca = 2
nuca = 2
n = 6
a = [0.0, 0.0, 7.0, 3.0, 1.0, 4.0, $
0.0, 5.0, 1.0, 2.0, 1.0, 2.0, 1.0, 2.0, 4.0, 6.0, 3.0, 1.0]
frobenius_norm = IMSL_MATRIX_NORM(n, nlca, nuca, a, /SYMMETRIC)
inf_norm = IMSL_MATRIX_NORM(n, nlca, nuca, a, /INF_NORM, $
/SYMMETRIC)
one_norm = IMSL_MATRIX_NORM(n, nlca, nuca, a, /ONE_NORM, $
/SYMMETRIC)
PRINT, 'Frobenius norm = ', frobenius_norm
PRINT, 'Infinity norm = ', inf_norm
PRINT, 'One norm = ', one_norm
Frobenius norm = 16.9411
Infinity norm = 16.0000
One norm = 16.0000
Example 4
Compute the Frobenius norm, infinity norm, and one norm of matrix a. Matrix a is stored in coordinate format.
nrows = 6
ncols = 6
a = REPLICATE(imsl_f_sp_elem, 15)
a(*).row = [0, 1, 1, 1, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5]
a(*).col = [0, 1, 2, 3, 2, 0, 3, 4, 0, 3, 4, 5, 0, 1, 5]
a(*).val = [10.0, 10.0, -3.0, -1.0, 15.0, $
-2.0, 10.0, -1.0, -1.0, -5.0, 1.0, -3.0, -1.0, -2.0, 6.0]
frobenius_norm = IMSL_MATRIX_NORM(nrows, ncols, a)
inf_norm = IMSL_MATRIX_NORM(nrows, ncols, a, /INF_NORM)
one_norm = IMSL_MATRIX_NORM(nrows, ncols, a, /ONE_NORM)
PRINT, 'Frobenius norm = ', frobenius_norm
PRINT, 'Infinity norm = ', inf_norm
PRINT, 'One norm = ', one_norm
Frobenius norm = 24.8395
Infinity norm = 15.0000
One norm = 18.0000
Example 5
Compute the Frobenius norm, infinity norm and one norm of matrix a. Matrix a is stored in symmetric coordinate format.
nrows = 6
ncols = 6
a = REPLICATE(imsl_f_sp_elem, 9)
a(*).row = [0, 0, 0, 1, 1, 2, 2, 4, 4]
a(*).col = [0, 2, 5, 3, 4, 2, 5, 4, 5]
a(*).val = [10.0, -1.0, 5.0, 2.0, 3.0, 3.0, 4.0, -1.0, 4.0]
frobenius_norm = IMSL_MATRIX_NORM(nrows, ncols, a, /SYMMETRIC)
inf_norm = IMSL_MATRIX_NORM(nrows, ncols, a, /INF_NORM, $
/SYMMETRIC)
one_norm = IMSL_MATRIX_NORM(nrows, ncols, a, /ONE_NORM, $
/SYMMETRIC)
PRINT, 'Frobenius norm = ', frobenius_norm
PRINT, 'Infinity norm = ', inf_norm
PRINT, 'One norm = ', one_norm
Frobenius norm = 15.8745
Infinity norm = 16.0000
One norm = 16.0000
Syntax
To compute various norms of a rectangular matrix:
Result = IMSL_MATRIX_NORM(A [, /DOUBLE] [, INF_NORM=value] [, ONE_NORM=value] [, SYMMETRIC=value])
To compute various norms of a matrix stored in band format:
Result = IMSL_MATRIX_NORM(Nn, Nlca, Nuca, A [, /DOUBLE] [, INF_NORM=value] [, ONE_NORM=value] [, SYMMETRIC=value])
To compute various norms of a matrix stored in coordinate format:
Result = IMSL_MATRIX_NORM(Nrows, Ncols, A [, /DOUBLE] [, INF_NORM=value] [, ONE_NORM=value] [, SYMMETRIC=value])
Return Value
The requested norm of the input matrix, by default, the Frobenius norm. If the norm cannot be computed, NaN is returned.
Arguments
A
Matrix for which the norm will be computed.
N
The order of matrix a.
Ncols
The number of columns in matrix a.
Nlca
Number of lower codiagonals of a.
Nrows
The number of rows in matrix a.
Nuca
Number of upper co-diagonals of a.
Keywords
DOUBLE (optional)
If present and nonzero, double precision is used.
INF_NORM (optional)
If present and nonzero, IMSL_MATRIX_NORM computes the infinity norm of matrix a.
ONE_NORM (optional)
If present and nonzero, IMSL_MATRIX_NORM computes the one norm of matrix a.
SYMMETRIC (optional)
If present and nonzero, matrix a is stored in symmetric storage mode. Keyword SYMMETRIC can not be used with a rectangular matrix.
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