NAG CL Interface
f01epc (real_tri_matrix_sqrt)
1
Purpose
f01epc computes the principal matrix square root, ${A}^{1/2}$, of a real upper quasitriangular $n$ by $n$ matrix $A$.
2
Specification
void 
f01epc (Integer n,
double a[],
Integer pda,
NagError *fail) 

The function may be called by the names: f01epc or nag_matop_real_tri_matrix_sqrt.
3
Description
A square root of a matrix $A$ is a solution $X$ to the equation ${X}^{2}=A$. A nonsingular matrix has multiple square roots. For a matrix with no eigenvalues on the closed negative real line, the principal square root, denoted by ${A}^{1/2}$, is the unique square root whose eigenvalues lie in the open right halfplane.
f01epc computes
${A}^{1/2}$, where
$A$ is an upper quasitriangular matrix, with
$1\times 1$ and
$2\times 2$ blocks on the diagonal. Such matrices arise from the Schur factorization of a real general matrix, as computed by
f08pec, for example.
f01epc does not require
$A$ to be in the canonical Schur form described in
f08pec, it merely requires
$A$ to be upper quasitriangular.
${A}^{1/2}$ then has the same block triangular structure as
$A$.
The algorithm used by
f01epc is described in
Higham (1987). In addition a blocking scheme described in
Deadman et al. (2013) is used.
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 $\left(i,j\right)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[\left(j1\right)\times {\mathbf{pda}}+i1\right]$.
On entry: the $n$ by $n$ upper quasitriangular matrix $A$.
On exit: the $n$ by $n$ principal matrix square root ${A}^{1/2}$.

3:
$\mathbf{pda}$ – Integer
Input

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

4:
$\mathbf{fail}$ – 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_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 $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_EIGENVALUES

$A$ has negative or vanishing eigenvalues. The principal square root is not defined in this case.
f01enc or
f01fnc may be able to provide further information.
 NE_INT

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 0$.
 NE_INT_2

On entry, ${\mathbf{pda}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
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_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.
7
Accuracy
The computed square root $\hat{X}$ satisfies ${\hat{X}}^{2}=A+\Delta A$, where ${\Vert \Delta A\Vert}_{F}\approx O\left(\epsilon \right)n{\Vert \hat{X}\Vert}_{F}^{2}$, where $\epsilon $ is machine precision.
8
Parallelism and Performance
f01epc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01epc 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.
The cost of the algorithm is
${n}^{3}/3$ floatingpoint operations; see Algorithm 6.7 of
Higham (2008).
$O\left(n\right)$ of integer allocatable memory is required by the function.
If
$A$ is a full matrix, then
f01enc should be used to compute the square root. If
$A$ has negative real eigenvalues then
f01fnc can be used to return a complex, nonprincipal square root.
If condition number and residual bound estimates are required, then
f01jdc should be used. For further discussion of the condition of the matrix square root see Section 6.1 of
Higham (2008).
10
Example
This example finds the principal matrix square root of the matrix
10.1
Program Text
10.2
Program Data
10.3
Program Results