. 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.

nag_matop_real_tri_matrix_sqrt (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 nag_dhseqr (f08pec), for example. nag_matop_real_tri_matrix_sqrt (f01epc) does not require

A

to be in the canonical Schur form described in nag_dhseqr (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 nag_matop_real_tri_matrix_sqrt (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: $n$ – IntegerInput: On entry: $n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $a [\dim]$ – doubleInput/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 $n$ by $n$ upper quasi-triangular matrix $A$ .

On exit: the $n$ by $n$ principal matrix square root $A^{1 / 2}$ .
3: $pda$ – IntegerInput: 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 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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. nag_matop_real_gen_matrix_sqrt (f01enc) or nag_matop_complex_gen_matrix_sqrt (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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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

nag_matop_real_tri_matrix_sqrt (f01epc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_matop_real_tri_matrix_sqrt (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 nag_matop_real_gen_matrix_sqrt (f01enc) should be used to compute the square root. If

A

has negative real eigenvalues then nag_matop_complex_gen_matrix_sqrt (f01fnc) can be used to return a complex, non-principal square root.

If condition number and residual bound estimates are required, then nag_matop_real_gen_matrix_cond_sqrt (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}) .

NAG Library Function Document

nag_matop_real_tri_matrix_sqrt (f01epc)

▸▿ 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

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