NAG CL Interface
f01jdc (real_gen_matrix_cond_sqrt)
1
Purpose
f01jdc 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.
2
Specification
The function may be called by the names: f01jdc or nag_matop_real_gen_matrix_cond_sqrt.
3
Description
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 halfplane.
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
The absolute condition number is given by the norm of the Fréchet derivative which is defined by
The Fréchet derivative is linear in
$E$ and can, therefore, be written as
where the
$\mathrm{vec}$ operator stacks the columns of a matrix into one vector, so that
$K\left(A\right)$ is
${n}^{2}\times {n}^{2}$.
f01jdc uses Algorithm 3.20 from
Higham (2008) to compute an estimate
$\gamma $ such that
$\gamma \le {\Vert K\left(X\right)\Vert}_{F}$. The quantity of
$\gamma $ provides a good approximation to
${\Vert L\left(A\right)\Vert}_{F}$. The relative condition number,
${\kappa}_{{A}^{1/2}}$, is then computed via
${\kappa}_{{A}^{1/2}}$ 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
Section 7). It is computed via
4
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
5
Arguments

1:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.

2:
$\mathbf{a}\left[\mathit{dim}\right]$ – double
Input/Output

Note: the dimension,
dim, of the array
a
must be at least
${\mathbf{pda}}\times {\mathbf{n}}$.
The $(i,j)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[(j1)\times {\mathbf{pda}}+i1\right]$.
On entry: the $n\times n$ matrix $A$.
On exit: contains, if
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the
$n\times n$ principal matrix square root,
${A}^{1/2}$. Alternatively, if
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_EIGENVALUES, contains an
$n\times n$ nonprincipal square root of
$A$.

3:
$\mathbf{pda}$ – Integer
Input

On entry: the stride separating matrix row elements in the array
a.
Constraint:
${\mathbf{pda}}\ge {\mathbf{n}}$.

4:
$\mathbf{alpha}$ – double *
Output

On exit: an estimate of the stability of the relative residual for the computed principal (if
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR) or nonprincipal (if
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_EIGENVALUES) matrix square root,
$\alpha $.

5:
$\mathbf{condsa}$ – double *
Output

On exit: an estimate of the relative condition number, in the Frobenius norm, of the principal (if
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR) or nonprincipal (if
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_EIGENVALUES) matrix square root at
$A$,
${\kappa}_{{A}^{1/2}}$.

6:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ALG_FAIL

An error occurred when computing the condition number. The matrix square root was still returned but you should use
f01enc to check if it is the principal matrix square root.
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.
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
 NE_BAD_PARAM

On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
 NE_EIGENVALUES

$A$ has a semisimple vanishing eigenvalue. A nonprincipal square root was returned.
 NE_INT

On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 0$.
 NE_INT_2

On entry, ${\mathbf{pda}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
 NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
 NE_NEGATIVE_EIGVAL

$A$ has a negative real eigenvalue. The principal square root is not defined.
f01kdc can be used to return a complex, nonprincipal square root.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
 NE_SINGULAR

$A$ has a defective vanishing eigenvalue. The square root and condition number cannot be found in this case.
7
Accuracy
If the computed square root is
$\stackrel{~}{X}$, then the relative residual
is bounded approximately by
$n\alpha \epsilon $, where
$\epsilon $ is
machine precision. The relative error in
$\stackrel{~}{X}$ is bounded approximately by
$n\alpha {\kappa}_{{A}^{1/2}}\epsilon $.
8
Parallelism and Performance
Background information to multithreading can be found in the
Multithreading documentation.
f01jdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01jdc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
Approximately $3\times {n}^{2}$ of real allocatable memory is required by the function.
The cost of computing the matrix square root is $85{n}^{3}/3$ floatingpoint operations. The cost of computing the condition number depends on how fast the algorithm converges. It typically takes over twice as long as computing the matrix square root.
If condition estimates are not required then it is more efficient to use
f01enc to obtain the matrix square root alone. Condition estimates for the square root of a complex matrix can be obtained via
f01kdc.
10
Example
This example estimates the matrix square root and condition number of the matrix
10.1
Program Text
10.2
Program Data
10.3
Program Results