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Reciprocal condition number




C = rcond(A) returns an estimate for the reciprocal condition of A in 1-norm. If A is well conditioned, rcond(A) is near 1.0. If A is badly conditioned, rcond(A) is near 0.


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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 =


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.

ans =


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.

ans =


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.

ans =


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

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A — Input matrixsquare numeric matrix

Input matrix, specified as a square numeric matrix.

Data Types: single | double

Output Arguments

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C — Reciprocal condition numberscalar

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.

More About

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

See Also

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Introduced before R2006a

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