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

α

is a measure of the stability of the relative residual (see Section 7). It is computed via

α = \frac{{‖ A^{1 / 2} ‖}_{F}^{2}}{{‖ A ‖}_{F}} .

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: $n$ – Integer Input: On entry: $n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $a [\dim]$ – double Input/Output: Note: the dimension, dim, of the array a must be at least $pda \times n$ .

The $(i, j)$ th element of the matrix $A$ is stored in $a [(j - 1) \times pda + i - 1]$ .

On entry: the $n \times n$ matrix $A$ .

On exit: contains, if $fail . code =$ NE_NOERROR, the $n \times n$ principal matrix square root, $A^{1 / 2}$ . Alternatively, if $fail . code =$ NE_EIGENVALUES, contains an $n \times n$ non-principal square root of $A$ .
3: $pda$ – Integer Input: On entry: the stride separating matrix row elements in the array a.

Constraint: $pda \geq n$ .
4: $alpha$ – double * Output: On exit: an estimate of the stability of the relative residual for the computed principal (if $fail . code =$ NE_NOERROR) or non-principal (if $fail . code =$ NE_EIGENVALUES) matrix square root, $α$ .
5: $condsa$ – double * Output: On exit: an estimate of the relative condition number, in the Frobenius norm, of the principal (if $fail . code =$ NE_NOERROR) or non-principal (if $fail . code =$ NE_EIGENVALUES) matrix square root at $A$ , $κ_{A^{1 / 2}}$ .
6: $fail$ – 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 $⟨ value ⟩$ had an illegal value.
NE_EIGENVALUES: $A$ has a semisimple vanishing eigenvalue. A non-principal square root was returned.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pda \geq n$ .
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, non-principal 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

\tilde{X}

, then the relative residual

\frac{{‖ A - {\tilde{X}}^{2} ‖}_{F}}{{‖ A ‖}_{F}},

is bounded approximately by

n α ε

, where

ε

is machine precision. The relative error in

\tilde{X}

is bounded approximately by

n α κ_{A^{1 / 2}} ε

8 Parallelism and Performance

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 implementation-specific information.

9 Further Comments

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

floating-point 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

A = (\begin{array}{r} −5 & 2 & −1 & 1 \\ −2 & −3 & 19 & 27 \\ −9 & 0 & 15 & 24 \\ 7 & 8 & 11 & 16 \end{array}) .

f01jd: FL CL CPP AD

NAG CL Interfacef01jdc (real_​gen_​matrix_​cond_​sqrt)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
f01jdc (real_gen_matrix_cond_sqrt)