matrixNormBand

Computes various norms of a matrix stored in band storage mode.

Synopsis

matrixNormBand (n, a, nlc, nuc)

Required Arguments

int n (Input)

The order of matrix A.

float a[] (Input)

Matrix for which the norm will be computed.

int nlc (Input)

Number of lower codiagonals of A.

int nuc (Input)

Number of upper codiagonals of A.

Return Value

The requested norm of the input matrix, by default, the Frobenius norm. If the norm cannot be computed, NaN is returned.

Optional Arguments

oneNorm,

Compute the 1-norm of matrix A,

infNorm,

Compute the infinity norm of matrix A,

symmetric,

Matrix A is stored in band symmetric storage mode.

Description

By default, matrixNormBand computes the Frobenius norm

\[\|A\|_2 = \left[\sum_{i=0}^{m-1} \sum_{j=0}^{n-1} A_{ij}^2\right]^{\frac{1}{2}}\]

If the option oneNorm is selected, the 1-norm

\[\|A\|_1 = \max_{0 \leq j \leq n-1} \sum_{i=0}^{m-1} \left|A_{ij}\right|\]

is returned. If the option infNorm is selected, the infinity norm

\[\|A\|_{\infty} = \max_{0 \leq i \leq m-1} \sum_{j=0}^{n-1} \left|A_{ij}\right|\]

is returned.

Examples

Example 1

Compute the Frobenius norm, infinity norm, and one norm of matrix A. Matrix A is stored in band storage mode.

from __future__ import print_function
from pyimsl.math.matrixNormBand import matrixNormBand

a = [[0., 2., 3., -1.],
     [1., 1., 1., 1.],
     [0., 3., 4., 0.]]
nlc = 1
nuc = 1
n = 4

frobenius_norm = matrixNormBand(n, a, nlc, nuc)

inf_norm = matrixNormBand(n, a, nlc, nuc, infNorm=True)

one_norm = matrixNormBand(n, a, nlc, nuc, oneNorm=True)

print("Frobenius norm = %10.6f" % frobenius_norm)
print("Infinity norm  = %10.6f" % inf_norm)
print("One norm       = %10.6f" % one_norm)

Output

Frobenius norm =   6.557439
Infinity norm  =   5.000000
One norm       =   8.000000

Example 2

Compute the Frobenius norm, infinity norm, and one norm of matrix A. Matrix A is stored in symmetric band storage mode.

from __future__ import print_function
from pyimsl.math.matrixNormBand import matrixNormBand

a = [[0., 0., 7., 3., 1., 4.],
     [0., 5., 1., 2., 1., 2.],
     [1., 2., 4., 6., 3., 1.]]
nlc = 2
nuc = 2
n = 6

frobenius_norm = matrixNormBand(n, a, nlc, nuc, symmetric=True)

inf_norm = matrixNormBand(n, a, nlc, nuc, infNorm=True, symmetric=True)

one_norm = matrixNormBand(n, a, nlc, nuc, oneNorm=True, symmetric=True)

print("Frobenius norm = %10.6f" % frobenius_norm)
print("Infinity norm  = %10.6f" % inf_norm)
print("One norm       = %10.6f" % one_norm)

Output

Frobenius norm =  16.941074
Infinity norm  =  16.000000
One norm       =  16.000000