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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_matop_complex_gen_matrix_sqrt (f01fn)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_matop_complex_gen_matrix_sqrt (f01fn) computes the principal matrix square root, A1/2, of a complex n by n matrix A.


[a, ifail] = f01fn(a, 'n', n)
[a, ifail] = nag_matop_complex_gen_matrix_sqrt(a, 'n', n)


A square root of a matrix A is a solution X to the equation X2=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 A1/2, is the unique square root whose eigenvalues lie in the open right half-plane.
A1/2 is computed using the algorithm described in Björck and Hammarling (1983). In addition a blocking scheme described in Deadman et al. (2013) is used.


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 (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA


Compulsory Input Parameters

1:     alda: – complex array
The first dimension of the array a must be at least n.
The second dimension of the array a must be at least n.
The n by n matrix A.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array a.
n, the order of the matrix A.
Constraint: n0.

Output Parameters

1:     alda: – complex array
The first dimension of the array a will be n.
The second dimension of the array a will be n.
Contains, if ifail=0, the n by n principal matrix square root, A1/2. Alternatively, if ifail=1, contains an n by n non-principal square root of A.
2:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
A has a negative or semisimple vanishing eigenvalue. A non-principal square root is returned.
A has a defective vanishing eigenvalue. The square root cannot be found in this case.
An internal error occurred. It is likely that the function was called incorrectly.
Constraint: n0.
Constraint: ldan.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


The computed square root X^ satisfies X^2=A+ΔA, where ΔAFOεn3X^F2, where ε is machine precision.

Further Comments

The cost of the algorithm is 85n3/3 complex floating-point operations; see Algorithm 6.3 in Higham (2008). O2×n2 of complex allocatable memory is required by the function.
If condition number and residual bound estimates are required, then nag_matop_complex_gen_matrix_cond_sqrt (f01kd) should be used. For further discussion of the condition of the matrix square root see Section 6.1 of Higham (2008).


This example finds the principal matrix square root of the matrix
A = 105+121i -21+157i 42+18i -4-02i 174+072i 28+236i 51+31i 16-06i 176+052i 37+177i 23+27i 25+13i -9+125i -111+067i -8+30i 08i .  
function f01fn_example

fprintf('f01fn example results\n\n');

% Principal square root of complex matrix A

a = [ 105 + 121i   -21 + 157i   42 + 18i   -4 -  2i;
      174 +  72i    28 + 236i   51 + 31i   16 -  6i;
      176 +  52i    37 + 177i   23 + 27i   25 + 13i;
       -9 + 125i  -111 +  67i   -8 + 30i         8i]; 

[as, ifail] = f01fn(a);

disp('Square root of A:');

f01fn example results

Square root of A:
  10.0000 + 5.0000i   3.0000 + 6.0000i   2.0000 - 1.0000i  -1.0000 + 1.0000i
   7.0000 - 1.0000i   9.0000 +10.0000i   3.0000 + 0.0000i  -0.0000 - 1.0000i
   7.0000 - 4.0000i   5.0000 + 5.0000i   3.0000 + 3.0000i   4.0000 - 1.0000i
   2.0000 + 5.0000i  -2.0000 + 5.0000i  -1.0000 + 2.0000i   2.0000 - 0.0000i

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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