. 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 half-plane.

f01epc computes

A^{1 / 2}

, where

A

is an upper quasi-triangular 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 quasi-triangular.

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: $n$ – 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 $n \times n$ upper quasi-triangular matrix $A$ .

On exit: the $n \times n$ principal matrix square root $A^{1 / 2}$ .
3: $pda$ – Integer Input: On entry: the stride separating matrix row elements in the array a.

Constraint: $pda \geq n$ .
4: $fail$ – 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 $⟨ value ⟩$ 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, $n = ⟨ value ⟩$ .
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_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 + Δ A

, where

{‖ Δ A ‖}_{F} \approx O (ε) n {‖ \hat{X} ‖}_{F}^{2}

, where

ε

is machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

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 implementation-specific information.

9 Further Comments

The cost of the algorithm is

n^{3} / 3

floating-point operations; see Algorithm 6.7 of Higham (2008).

O (n)

of integer allocatable memory is required by the function.

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, non-principal 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

A = (\begin{array}{r} 6 & 4 & −5 & 15 \\ 8 & 6 & −3 & 10 \\ 0 & 0 & 3 & −4 \\ 0 & 0 & 4 & 3 \end{array}) .

f01ep: FL CL CPP AD PY MB

NAG CL Interfacef01epc (real_​tri_​matrix_​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
f01epc (real_tri_matrix_sqrt)