# NAG Library Function Document

## 1Purpose

nag_matop_real_gen_matrix_cond_sqrt (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 $n$ by $n$ matrix $A$. The principal square root, ${A}^{1/2}$, of $A$ is also returned.

## 2Specification

 #include #include
 void nag_matop_real_gen_matrix_cond_sqrt (Integer n, double a[], Integer pda, double *alpha, double *condsa, NagError *fail)

## 3Description

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 half-plane.
The Fréchet derivative of a matrix function ${A}^{1/2}$ in the direction of the matrix $E$ is the linear function mapping $E$ to $L\left(A,E\right)$ such that
 $A+E1/2 - A1/2 - LA,E = oA .$
The absolute condition number is given by the norm of the Fréchet derivative which is defined by
 $LA := maxE≠0 LA,E E .$
The Fréchet derivative is linear in $E$ and can therefore be written as
 $vec LA,E = KA vecE ,$
where the $\mathrm{vec}$ operator stacks the columns of a matrix into one vector, so that $K\left(A\right)$ is ${n}^{2}×{n}^{2}$.
nag_matop_real_gen_matrix_cond_sqrt (f01jdc) uses Algorithm 3.20 from Higham (2008) to compute an estimate $\gamma$ such that $\gamma \le {‖K\left(X\right)‖}_{F}$. The quantity of $\gamma$ provides a good approximation to ${‖L\left(A\right)‖}_{F}$. The relative condition number, ${\kappa }_{{A}^{1/2}}$, is then computed via
 $κA1/2 = LAF AF A1/2 F .$
${\kappa }_{{A}^{1/2}}$ 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
 $α= A 1/2 F 2 AF .$

## 4References

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

## 5Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:    $\mathbf{a}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array a must be at least ${\mathbf{pda}}×{\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$.
On entry: the $n$ by $n$ matrix $A$.
On exit: contains, if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the $n$ by $n$ principal matrix square root, ${A}^{1/2}$. Alternatively, if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_EIGENVALUES, contains an $n$ by $n$ non-principal square root of $A$.
3:    $\mathbf{pda}$IntegerInput
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 non-principal (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 non-principal (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 3.7 in How to Use the NAG Library and its Documentation).

## 6Error 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 nag_matop_real_gen_matrix_sqrt (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 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_EIGENVALUES
$A$ has a semisimple vanishing eigenvalue. A non-principal square root was returned.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INT_2
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NEGATIVE_EIGVAL
$A$ has a negative real eigenvalue. The principal square root is not defined. nag_matop_complex_gen_matrix_cond_sqrt (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 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_SINGULAR
$A$ has a defective vanishing eigenvalue. The square root and condition number cannot be found in this case.

## 7Accuracy

If the computed square root is $\stackrel{~}{X}$, then the relative residual
 $A - X~2 F AF ,$
is bounded approximately by $n\alpha \epsilon$, where $\epsilon$ is machine precision. The relative error in $\stackrel{~}{X}$ is bounded approximately by $n\alpha {\kappa }_{{A}^{1/2}}\epsilon$.

## 8Parallelism and Performance

nag_matop_real_gen_matrix_cond_sqrt (f01jdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_matop_real_gen_matrix_cond_sqrt (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.

Approximately $3×{n}^{2}$ 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 nag_matop_real_gen_matrix_sqrt (f01enc) to obtain the matrix square root alone. Condition estimates for the square root of a complex matrix can be obtained via nag_matop_complex_gen_matrix_cond_sqrt (f01kdc).

## 10Example

This example estimates the matrix square root and condition number of the matrix
 $A = -5 2 -1 1 -2 -3 19 27 -9 0 15 24 7 8 11 16 .$

### 10.1Program Text

Program Text (f01jdce.c)

### 10.2Program Data

Program Data (f01jdce.d)

### 10.3Program Results

Program Results (f01jdce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017