vectorNorm

Computes various norms of a vector or the difference of two vectors.

Synopsis

vectorNorm (x)

Required Arguments

float x (Input)

Input vector for which the norm is to be computed

Return Value

The requested norm of the input vector. If the norm cannot be computed, NaN is returned.

Optional Arguments

oneNorm

Return the 1-norm.

infNorm (Output)

Compute the infinity norm,

\[\|x\|_{\infty} = \max_{0 \leq i < n} \left|x_i\right|\]

The index at which the vector has its maximum absolute value is also returned.

secondVector (Input)

Return the norm of x-y.

Description

By default, vectorNorm computes the Euclidean norm

\[\left(\sum_{i=0}^{n-1} x_i^2\right)^{\tfrac{1}{2}}\]

If the option oneNorm is selected, the 1-norm

\[\sum_{i=0}^{n-1} \left|x_i\right|\]

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

\[\max ∣x_i∣\]

is returned. In the case of the infinity norm, the program also returns the index of the element with maximum modulus. If secondVector is selected, then the norm of x −y is computed.

Examples

Example 1

In this example, the Euclidean norm of an input vector is computed.

from __future__ import print_function
from pyimsl.math.vectorNorm import vectorNorm

x = [1., 3., -2., 4.]
norm = vectorNorm(x)

print("Euclidian norm of x = ", norm)

Output

Euclidian norm of x =  5.477225575051661

Example 2

This example computes \(\max|x_i-y_i|\) and prints the norm and index.

from __future__ import print_function
from pyimsl.math.vectorNorm import vectorNorm

x = [1., 3., -2., 4.]
y = [4., 2., -1., -5.]
index = []

norm = vectorNorm(x, secondVector=y, infNorm=index)

print("Infinity norm of x-y =", norm, end=' ')
print("at location", index[0])

Output

Infinity norm of x-y = 9.0 at location 3