Documentation

cond

Condition number for inversion

Description

example

C = cond(A) returns the 2-norm condition number for inversion, equal to the ratio of the largest singular value of A to the smallest.

example

C = cond(A,p) returns the p-norm condition number, where p can be 1, 2, Inf, or 'fro'.

Examples

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Calculate the condition number of a matrix and examine the sensitivity to the inverse calculation.

Create a 2-by-2 matrix.

A = [4.1 2.8;
9.7 6.6];

Calculate the 2-norm condition number of A.

C = cond(A)
C = 1.6230e+03

Since the condition number of A is much larger than 1, the matrix is sensitive to the inverse calculation. Calculate the inverse of A, and then make a small change in the second row of A and calculate the inverse again.

invA = inv(A)
invA = 2×2

-66.0000   28.0000
97.0000  -41.0000

A2 = [4.1    2.8;
9.671  6.608]
A2 = 2×2

4.1000    2.8000
9.6710    6.6080

invA2 = inv(A2)
invA2 = 2×2

472.0000 -200.0000
-690.7857  292.8571

The results indicate that making a small change in A can completely change the result of the inverse calculation.

Calculate the 1-norm condition number of a matrix.

Create a 3-by-3 matrix.

A = [1 0 -2;
3 4  6;
-1 5  7];

Calculate the 1-norm condition number of A. The value of the 1-norm condition number for an m-by-n matrix is

${\kappa }_{1}\left(A\right)=||A|{|}_{1}\phantom{\rule{0.16666666666666666em}{0ex}}\phantom{\rule{0.16666666666666666em}{0ex}}\phantom{\rule{0.16666666666666666em}{0ex}}||{A}^{-1}|{|}_{1}$,

where the 1-norm is the maximum absolute column sum of the matrix given by

$||A|{|}_{1}=\underset{1\le j\le n}{\text{max}}\sum _{i=1}^{m}|{a}_{ij}|.$

C = cond(A,1)
C = 18.0000

For this matrix the condition number is not too large, so the matrix is not particularly sensitive to the inverse calculation.

Input Arguments

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Input matrix. A can be either square or rectangular in size.

Data Types: single | double
Complex Number Support: Yes

Norm type, specified as one of the values shown in this table. cond computes the condition number using norm(A,p) * norm(inv(A),p) for values of p other than 2. See the norm page for additional information about these norm types.

Value of p

Norm Type

1

1-norm condition number

2

2-norm condition number

Inf

Infinity norm condition number

'fro'

Frobenius norm condition number

Example: cond(A,1) calculates the 1-norm condition number.

Output Arguments

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Condition number, returned as a scalar. Values of C near 1 indicate a well-conditioned matrix, and large values of C indicate an ill-conditioned matrix. Singular matrices have a condition number of Inf.

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Condition Number for Inversion

A condition number for a matrix and computational task measures how sensitive the answer is to changes in the input data and roundoff errors in the solution process.

The condition number for inversion of a matrix measures the sensitivity of the solution of a system of linear equations to errors in the data. It gives an indication of the accuracy of the results from matrix inversion and the linear equation solution. For example, the 2-norm condition number of a matrix is

$\kappa \left(A\right)=‖{A}^{}‖‖{A}^{-1}‖\text{\hspace{0.17em}}.$

In this context, a large condition number indicates that a small change in the coefficient matrix A can lead to larger changes in the output b in the linear equations Ax = b and xA = b. The extreme case is when A is so poorly conditioned that it is singular (an infinite condition number), in which case it has no inverse and the linear equation has no unique solution.

Tips

• rcond is a more efficient, but less reliable, method of estimating the condition of a matrix compared to cond.

Algorithms

The algorithm for cond has three pieces:

• If p = 2, then cond uses the singular value decomposition provided by svd to find the ratio of the largest and smallest singular values.

• If p = 1, Inf, or 'fro', then cond calculates the condition number using the appropriate norm of the input matrix and its inverse with norm(A,p) * norm(inv(A),p).

• If the input matrix is sparse, then cond ignores any specified p value and calls condest.