fixed.realConditionNumberUpperBound

Estimate of upper bound for 2-norm condition number of real-valued matrix

Since R2022b

Syntax

C = fixed.realConditionNumberUpperBound(m,n,max_abs_A,noiseStandardDeviation)

C = fixed.realConditionNumberUpperBound(___,p_s)

C = fixed.realConditionNumberUpperBound(___,regularizationParameter)

Description

C = fixed.realConditionNumberUpperBound(m,n,max_abs_A,noiseStandardDeviation) returns an estimate of an upper bound for the 2-norm condition number of a real-valued m-by-n matrix A, where max_abs_A >= max(abs(A(:))) and noiseStandardDeviation is the standard deviation of the additive random noise in A.

C = fixed.realConditionNumberUpperBound(___,p_s) uses the probability p_s that the estimate of the lower bound of the smallest singular value is larger than the actual smallest singular value. p_s is an optional parameter. If not supplied or empty, then the default value is used.

C = fixed.realConditionNumberUpperBound(___,regularizationParameter) returns an estimate of an upper bound for the 2-norm condition number of a real-valued matrix $$\left[\begin{array}{c}\lambda I_n\\ A\end{array}\right]$$, where λ is the regularizationParameter, A is an m-by-n matrix with m >= n, and I_n = eye(n). regularizationParameter is an optional parameter. If not supplied or empty, then the default value is used.

Examples

Condition Number of Low Rank Matrix with Additive Noise

Estimate an upper bound for the 2-norm condition number of a low rank, real-valued matrix with additive noise.

Define a real matrix A with additive noise.

m = 300;
n = 10;
rankA = 3;
A = fixed.example.realRandomLowRankMatrix(m,n,rankA);

noiseStandardDeviation = sqrt(10^(-50/10));
A = A + fixed.example.realNormalRandomArray(0,...
    noiseStandardDeviation,m,n);

Estimate an upper bound for the condition number of the matrix A.

C = fixed.realConditionNumberUpperBound(m,n,...
    max(abs(A(:))),noiseStandardDeviation)

C =

   1.4780e+03

Compare to the actual condition number of the matrix.

C_actual = cond(A)

C_actual =

  275.5870

Condition Number of Low Rank Matrix with Regularization Parameter

Estimate an upper bound for the 2-norm condition number of a low rank, real-valued matrix with additive noise, using the regularization parameter.

Define a real matrix A with additive noise.

m = 300;
n = 10;
rankA = 3;
A = fixed.example.realRandomLowRankMatrix(m,n,rankA);

noiseStandardDeviation = sqrt(10^(-50/10));
A = A + fixed.example.realNormalRandomArray(0,...
    noiseStandardDeviation,m,n);

Define the regularization parameter.

regularizationParameter = 0.01;
A = [regularizationParameter*eye(n);A];

Estimate an upper bound for the condition number of the matrix A with the regularization parameter. Use the default value for p_s.

C = fixed.realConditionNumberUpperBound(m,n,...
    max(abs(A(:))),noiseStandardDeviation,[],...
    regularizationParameter)

C =

   1.4343e+03

Compare to the actual condition number of the matrix.

C_actual = cond(A)

C_actual =

  293.4647

Condition Number of Full Rank Matrix

Estimate an upper bound for the 2-norm condition number of a full rank random matrix with normally distributed elements.

Define a full rank, random, real matrix A with normally distributed elements.

m = 300;
n = 10;
noiseStandardDeviation = 1;
A = fixed.example.realNormalRandomArray(0,noiseStandardDeviation,m,n);

Estimate an upper bound for the condition number of the matrix A.

C = fixed.realConditionNumberUpperBound(m,n,...
    max(abs(A(:))),noiseStandardDeviation)

C =

   19.0850

Compare to the actual condition number of the matrix.

C_actual = cond(A)

C_actual =

    1.2801

Input Arguments

m — Number of rows in matrix A

positive integer-valued scalar

Number of rows in matrix A, specified as a positive integer-valued scalar. The number of rows, m, must be greater than or equal to the number of columns, n.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

n — Number of columns in matrix A

positive integer-valued scalar

Number of columns in matrix A, specified as a positive integer-valued scalar. The number of rows, m, must be greater than or equal to the number of columns, n.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

max_abs_A — Maximum of absolute value of matrix A

scalar

Maximum of absolute value of matrix A, specified as a scalar.

Example: max_abs_A >= max(abs(A(:)))

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

noiseStandardDeviation — Standard deviation of additive random noise in matrix A

(2^-precisionBits)/(sqrt(12)) (default) | scalar

Standard deviation of additive random noise in matrix A, specified as a scalar.

If noiseStandardDeviation is not supplied or empty, then the default value is used, which is the standard deviation of the quantization noise,

$${\sigma }_q=\frac{2^{-precisionBits}}{\sqrt{12}}.$$

This value is calculated by the function fixed.realQuantizationNoiseStandardDeviation.

If noiseStandardDeviation is zero, then fixed.singularValueLowerBound will return zero for the estimate of the smallest singular value, and fixed.realConditionNumberUpperBound will return an infinite condition number.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

p_s — Probability that estimate of lower bound of smallest singular value is larger than actual smallest singular value of matrix A

2.8665e-07 (default) | scalar

Probability that estimate of lower bound of smallest singular value is larger than actual smallest singular value of matrix A, specified as a scalar.

If p_s is not supplied or empty, then the default of p_s = (1/2)*(1+erf(-5/sqrt(2))) = 2.8665e-07 is used, which is five standard deviations below the mean. So, the probability that the estimated lower bound for the smallest singular value is less than the actual smallest singular value is 1 - p_s = 0.99999971 - p_s = 0.9999997.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

regularizationParameter — Regularization parameter

0 (default) | nonnegative scalar

Regularization parameter, specified as a nonnegative scalar. Small, positive values of the regularization parameter can improve the conditioning of the problem and reduce the variance of the estimates. While biased, the reduced variance of the estimate often results in a smaller mean squared error when compared to least-squares estimates.

regularizationParameter is the Tikhonov regularization parameter of the matrix [regularizationParameter*eye(n); A], where A is an m-by-n matrix with m >= n.

Data Types: single | double

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. The condition number for inversion gives an indication of the accuracy of the results from matrix inversion and the linear equation solution.

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 equation Ax = 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.

Algorithms

The condition number with respect to the inversion of matrix A is the ratio of the largest singular value of A to the smallest singular value of A. The fixed.realSingularValueLowerBound function estimates the lower bound of the smallest singular value, s_n, of A. The fixed.singularValueUpperBound function determines an upper bound for the largest singular value, svdUpperBound, of A. A bound on the condition number of A is then cond(A) = max(svd(A))/min(svd(A)) <= svdUpperBound/s_n [1][2][3].

References

[1] Bryan, Thomas A., Jenna L. Warren, Brenda Zhuang, and Jessica Clayton. Continuation in Part for "Systems and Methods for Design Parameter Selection." U.S. Patent Application No. 16/947, 130. 2022.

[2] Bryan, Thomas A. and Jenna L. Warren. "Systems and Methods for Design Parameter Selection." U.S. Patent Application No. 16/947, 130. 2020.

[3] Chen, Zizhong and Jack J. Dongarra. "Condition Numbers of Gaussian Random Matrices." SIAM Journal on Matrix Analysis and Applications 27, no. 3 (July 2005): 603-620. https://doi.org/10.1137/040616413.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced in R2022b