cond

Condition number for inversion

Syntax

C = cond(A)

C = cond(A,p)

Description

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.

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

Examples

Condition Number of Matrix

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.

1-Norm Condition Number

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

$${\kappa }_1\left(A\right)=||A|{|}_1||A^{-1}|{|}_1$$,

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

A — Input matrix

matrix

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

Data Types: single | double
Complex Number Support: Yes

p — Norm type

2 (default) | 1 | 'fro' | Inf

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

C — Condition number

scalar

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.

By convention, the condition number of an empty matrix is zero.

More About

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 square matrix is

$$\kappa \left(A\right)=\Vert A^{}\Vert \Vert A^{-1}\Vert .$$

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.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• cond does not support sparse matrices.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The cond function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray (Parallel Computing Toolbox). For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

Usage notes and limitations:

• cond does not support sparse matrices.

For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced before R2006a

R2021b: cond returns NaN for nonfinite inputs

cond returns NaN when the input contains nonfinite values (Inf or NaN). Previously, cond threw an error when the input contained nonfinite values.