naginterfaces.library.matop.real_​gen_​matrix_​cond_​sqrt

naginterfaces.library.matop.real_gen_matrix_cond_sqrt(a)

real_gen_matrix_cond_sqrt computes an estimate of the relative condition number ${\kappa }_{A^{1/2}}$ and a bound on the relative residual, in the Frobenius norm, for the square root of a real $n\times n$ matrix $A$. The principal square root, $A^{1/2}$, of $A$ is also returned.

For full information please refer to the NAG Library document for f01jd

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/f01/f01jdf.html

Parameters

afloat, array-like, shape $(n,n)$

The $n\times n$ matrix $A$.

Returns

afloat, ndarray, shape $(n,n)$

Contains, if no exception or warning is raised, the $n\times n$ principal matrix square root, $A^{1/2}$. Alternatively, if $errno$ = 1, contains an $n\times n$ non-principal square root of $A$.

alphafloat

An estimate of the stability of the relative residual for the computed principal (if no exception or warning is raised) or non-principal (if $errno$ = 1) matrix square root, $\alpha$.

condsafloat

An estimate of the relative condition number, in the Frobenius norm, of the principal (if no exception or warning is raised) or non-principal (if $errno$ = 1) matrix square root at $A$, ${\kappa }_{A^{1/2}}$.

Raises

NagValueError

(errno $-1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

(errno $2$)

$A$ has a defective vanishing eigenvalue. The square root and condition number cannot be found in this case.

(errno $3$)

$A$ has a negative real eigenvalue. The principal square root is not defined. complex_gen_matrix_cond_sqrt() can be used to return a complex, non-principal square root.

(errno $4$)

An error occurred when computing the matrix square root. Consequently, $alpha$ and $condsa$ could not be computed. It is likely that the function was called incorrectly.

Warns

NagAlgorithmicWarning

(errno $1$)

$A$ has a semisimple vanishing eigenvalue. A non-principal square root was returned.

(errno $5$)

An error occurred when computing the condition number. The matrix square root was still returned but you should use real_gen_matrix_sqrt() to check if it is the principal matrix square root.

Notes

For a matrix with no eigenvalues on the closed negative real line, the principal matrix square root, $A^{1/2}$, of $A$ is the unique square root with eigenvalues in the right half-plane.

The Fréchet derivative of a matrix function $A^{1/2}$ in the direction of the matrix $E$ is the linear function mapping $E$ to $L(A,E)$ such that

$${(A+E)}^{1/2}-A^{1/2}-L(A,E)=o\left(\parallel A\parallel \right)\text{.}$$

The absolute condition number is given by the norm of the Fréchet derivative which is defined by

$$\parallel L\left(A\right)\parallel :={max}_{E\ne 0}\frac{\parallel L(A,E)\parallel }{\parallel E\parallel }\text{.}$$

The Fréchet derivative is linear in $E$ and can, therefore, be written as

$$vec\left(L(A,E)\right)=K\left(A\right)vec\left(E\right)\text{,}$$

where the $vec$ operator stacks the columns of a matrix into one vector, so that $K\left(A\right)$ is $n^2\times n^2$.

real_gen_matrix_cond_sqrt uses Algorithm 3.20 from Higham (2008) to compute an estimate $\gamma$ such that $\gamma \le {\parallel K\left(X\right)\parallel }_F$. The quantity of $\gamma$ provides a good approximation to ${\parallel L\left(A\right)\parallel }_F$. The relative condition number, ${\kappa }_{A^{1/2}}$, is then computed via

$${\kappa }_{A^{1/2}}=\frac{{\parallel L\left(A\right)\parallel }_F{\parallel A\parallel }_F}{{\Vert A^{1/2}\Vert }_F}\text{.}$$

${\kappa }_{A^{1/2}}$ is returned in the argument $condsa$.

$A^{1/2}$ is computed using the algorithm described in Higham (1987). This is a real arithmetic version of the algorithm of Björck and Hammarling (1983). In addition, a blocking scheme described in Deadman et al. (2013) is used.

The computed quantity $\alpha$ is a measure of the stability of the relative residual (see Accuracy). It is computed via

$$\alpha =\frac{{\Vert A^{1/2}\Vert }_F^2}{{\parallel A\parallel }_F}\text{.}$$

References

Björck, Å and Hammarling, S, 1983, A Schur method for the square root of a matrix, Linear Algebra Appl. (52/53), 127–140

Deadman, E, Higham, N J and Ralha, R, 2013, Blocked Schur Algorithms for Computing the Matrix Square Root, Applied Parallel and Scientific Computing: 11th International Conference, (PARA 2012, Helsinki, Finland), P. Manninen and P. Öster, Eds, Lecture Notes in Computer Science (7782), 171–181, Springer–Verlag

Higham, N J, 1987, Computing real square roots of a real matrix, Linear Algebra Appl. (88/89), 405–430

Higham, N J, 2008, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, PA, USA