# rcond

Reciprocal condition number

## Syntax

## Description

## Examples

### Sensitivity of Badly Conditioned Matrix

Examine the sensitivity of a badly conditioned matrix.

A notable matrix that is symmetric and positive definite, but badly conditioned, is the Hilbert matrix. The elements of the Hilbert matrix are $$H(i,j)=1/(i+j-1)$$.

Create a 10-by-10 Hilbert matrix.

A = hilb(10);

Find the reciprocal condition number of the matrix.

C = rcond(A)

C = 2.8286e-14

The reciprocal condition number is small, so `A`

is badly conditioned.

The condition of `A`

has an effect on the solutions of similar linear systems of equations. To see this, compare the solution of $$Ax=b$$ to that of the perturbed system, $$Ax=b+0.01$$.

Create a column vector of ones and solve $$Ax=b$$.

b = ones(10,1); x = A\b;

Now change $$b$$ by `0.01`

and solve the perturbed system.

b1 = b + 0.01; x1 = A\b1;

Compare the solutions, `x`

and `x1`

.

norm(x-x1)

ans = 1.1250e+05

Since `A`

is badly conditioned, a small change in `b`

produces a very large change (on the order of 1e5) in the solution to `x = A\b`

. The system is sensitive to perturbations.

### Find Condition of Identity Matrix

Examine why the reciprocal condition number is a more accurate measure of singularity than the determinant.

Create a 5-by-5 multiple of the identity matrix.

A = eye(5)*0.01;

This matrix is full rank and has five equal singular values, which you can confirm by calculating `svd(A)`

.

Calculate the determinant of `A`

.

det(A)

ans = 1.0000e-10

Although the determinant of the matrix is close to zero, `A`

is actually very well conditioned and *not* close to being singular.

Calculate the reciprocal condition number of `A`

.

rcond(A)

ans = 1

The matrix has a reciprocal condition number of `1`

and is, therefore, very well conditioned. Use `rcond(A)`

or `cond(A)`

rather than `det(A)`

to confirm singularity of a matrix.

## Input Arguments

`A`

— Input matrix

square numeric matrix

Input matrix, specified as a square numeric matrix.

**Data Types: **`single`

| `double`

## Output Arguments

`C`

— Reciprocal condition number

scalar

Reciprocal condition number, returned as a scalar. The data
type of `C`

is the same as `A`

.

The reciprocal condition number is a scale-invariant measure of how close a given matrix is to the set of singular matrices.

If

`C`

is near 0, the matrix is nearly singular and badly conditioned.If

`C`

is near 1.0, the matrix is well conditioned.

## Tips

`rcond`

is a more efficient but less reliable method of estimating the condition of a matrix compared to the condition number,`cond`

.

## Extended Capabilities

### C/C++ Code Generation

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

Usage notes and limitations:

Code generation does not support sparse matrix inputs for this function.

### GPU Code Generation

Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Usage notes and limitations:

Code generation does not support sparse matrix inputs for this function.

### 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.

**Introduced before R2006a**

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