The function may be called by the names: f01enc or nag_matop_real_gen_matrix_sqrt.
3Description
A square root of a matrix is a solution to the equation . 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 , is the unique square root whose eigenvalues lie in the open right half-plane.
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 Science7782 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: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
2: – doubleInput/Output
Note: the dimension, dim, of the array a
must be at least
.
The th element of the matrix is stored in .
On entry: the matrix .
On exit: contains, if NE_NOERROR, the principal matrix square root, . Alternatively, if NE_EIGENVALUES, contains an non-principal square root of .
3: – IntegerInput
On entry: the stride separating matrix row elements in the array a.
Constraint:
.
4: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error 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 had an illegal value.
NE_EIGENVALUES
has a semisimple vanishing eigenvalue. A non-principal square root is returned.
NE_INT
On entry, .
Constraint: .
NE_INT_2
On entry, and .
Constraint: .
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
has a negative real eigenvalue. The principal square root is not defined. f01fnc 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
has a defective vanishing eigenvalue. The square root cannot be found in this case.
7Accuracy
The computed square root satisfies , where , where is machine precision.
For further discussion of the condition of the matrix square root see Section 6.1 of Higham (2008).
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f01enc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01enc 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.
9Further Comments
The cost of the algorithm is floating-point operations; see Algorithm 6.7 of Higham (2008). of real allocatable memory is required by the function.
If condition number and residual bound estimates are required, then f01jdc should be used.
10Example
This example finds the principal matrix square root of the matrix