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 .
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 to that of the perturbed system, .
Create a column vector of ones and solve .
b = ones(10,1); x = A\b;
Now change 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.
Version History
Introduced before R2006a
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