NAG Library Function Document

nag_zher2k (f16zrc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_zher2k (f16zrc) performs a rank-2k update on a complex Hermitian matrix.

2
Specification

#include <nag.h>
#include <nagf16.h>
void  nag_zher2k (Nag_OrderType order, Nag_UploType uplo, Nag_TransType trans, Integer n, Integer k, Complex alpha, const Complex a[], Integer pda, const Complex b[], Integer pdb, double beta, Complex c[], Integer pdc, NagError *fail)

3
Description

nag_zher2k (f16zrc) performs one of the Hermitian rank-2k update operations
CαABH + α-AH + βC   or   Cα-AHB + αBHA + β C ,  
where A and B are complex matrices, C is an n by n complex Hermitian matrix, α is a complex scalar, and β is a real scalar.

4
References

Basic Linear Algebra Subprograms Technical (BLAST) Forum (2001) Basic Linear Algebra Subprograms Technical (BLAST) Forum Standard University of Tennessee, Knoxville, Tennessee http://www.netlib.org/blas/blast-forum/blas-report.pdf

5
Arguments

1:     order Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     uplo Nag_UploTypeInput
On entry: specifies whether the upper or lower triangular part of C is stored.
uplo=Nag_Upper
The upper triangular part of C is stored.
uplo=Nag_Lower
The lower triangular part of C is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     trans Nag_TransTypeInput
On entry: specifies the operation to be performed.
trans=Nag_NoTrans
CαBHA + α-BAH + βC .
trans=Nag_ConjTrans
Cα-AHB + αBHA + βC .
Constraint: trans=Nag_NoTrans or Nag_ConjTrans.
4:     n IntegerInput
On entry: n, the order of the matrix C; the number of rows of A and B if trans=Nag_NoTrans, or the number of columns of A and B otherwise.
Constraint: n0.
5:     k IntegerInput
On entry: k, the number of columns of A and B if trans=Nag_NoTrans, or the number of rows of A and B otherwise.
Constraint: k0.
6:     alpha ComplexInput
On entry: the scalar α.
7:     a[dim] const ComplexInput
Note: the dimension, dim, of the array a must be at least
  • max1,pda×k when trans=Nag_NoTrans and order=Nag_ColMajor;
  • max1,n×pda when trans=Nag_NoTrans and order=Nag_RowMajor;
  • max1,pda×n when trans=Nag_ConjTrans and order=Nag_ColMajor;
  • max1,k×pda when trans=Nag_ConjTrans and order=Nag_RowMajor.
If order=Nag_ColMajor, Aij is stored in a[j-1×pda+i-1].
If order=Nag_RowMajor, Aij is stored in a[i-1×pda+j-1].
On entry: the matrix A; A is n by k if trans=Nag_NoTrans, or k by n otherwise.
8:     pda IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
  • if order=Nag_ColMajor,
    • if trans=Nag_NoTrans, pda max1,n ;
    • if trans=Nag_ConjTrans, pda max1,k ;
  • if order=Nag_RowMajor,
    • if trans=Nag_NoTrans, pdamax1,k;
    • if trans=Nag_ConjTrans, pdamax1,n.
9:     b[dim] const ComplexInput
Note: the dimension, dim, of the array b must be at least
  • max1,pdb×k when trans=Nag_NoTrans and order=Nag_ColMajor;
  • max1,n×pdb when trans=Nag_NoTrans and order=Nag_RowMajor;
  • max1,pdb×n when trans=Nag_ConjTrans and order=Nag_ColMajor;
  • max1,k×pdb when trans=Nag_ConjTrans and order=Nag_RowMajor.
If order=Nag_ColMajor, Bij is stored in b[j-1×pdb+i-1].
If order=Nag_RowMajor, Bij is stored in b[i-1×pdb+j-1].
On entry: the matrix B; B is n by k if trans=Nag_NoTrans, or k by n otherwise.
10:   pdb IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor,
    • if trans=Nag_NoTrans, pdb max1,n ;
    • if trans=Nag_ConjTrans, pdb max1,k ;
  • if order=Nag_RowMajor,
    • if trans=Nag_NoTrans, pdbmax1,k;
    • if trans=Nag_ConjTrans, pdbmax1,n.
11:   beta doubleInput
On entry: the scalar β.
12:   c[dim] ComplexInput/Output
Note: the dimension, dim, of the array c must be at least max1,pdc×n.
On entry: the n by n Hermitian matrix C.
If order=Nag_ColMajor, Cij is stored in c[j-1×pdc+i-1].
If order=Nag_RowMajor, Cij is stored in c[i-1×pdc+j-1].
If uplo=Nag_Upper, the upper triangular part of C must be stored and the elements of the array below the diagonal are not referenced.
If uplo=Nag_Lower, the lower triangular part of C must be stored and the elements of the array above the diagonal are not referenced.
On exit: the updated matrix C. The imaginary parts of the diagonal elements are set to zero.
13:   pdc IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix C in the array c.
Constraint: pdcmax1,n.
14:   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_ENUM_INT_2
On entry, trans=value, k=value, pda=value.
Constraint: if trans=Nag_ConjTrans, pda max1,k .
On entry, trans=value, k=value, pda=value.
Constraint: if trans=Nag_NoTrans, pdamax1,k.
On entry, trans=value, k=value, pdb=value.
Constraint: if trans=Nag_ConjTrans, pdb max1,k .
On entry, trans=value, k=value, pdb=value.
Constraint: if trans=Nag_NoTrans, pdbmax1,k.
On entry, trans=value, n=value, pda=value.
Constraint: if trans=Nag_ConjTrans, pdamax1,n.
On entry, trans=value, n=value, pda=value.
Constraint: if trans=Nag_NoTrans, pda max1,n .
On entry, trans=value, n=value, pdb=value.
Constraint: if trans=Nag_ConjTrans, pdbmax1,n.
On entry, trans=value, n=value, pdb=value.
Constraint: if trans=Nag_NoTrans, pdb max1,n .
NE_INT
On entry, k=value.
Constraint: k0.
On entry, n=value.
Constraint: n0.
NE_INT_2
On entry, pdc=value, n=value.
Constraint: pdcmax1,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.
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 BLAS standard requires accurate implementations which avoid unnecessary over/underflow (see Section 2.7 of Basic Linear Algebra Subprograms Technical (BLAST) Forum (2001)).

8
Parallelism and Performance

nag_zher2k (f16zrc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zher2k (f16zrc) 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

None.

10
Example

Perform rank-2k update of complex Hermitian 4 by 4 matrix C using 4 by 2 matrices A and B, C=-C+-0.5+0.5iABT+-0.5-0.5iBAT, where
C = 4.78+0.00i 2.00+0.30i 2.89+1.34i -1.89-1.15i 2.00-0.30i -4.11+0.00i 2.36+4.25i 0.04+3.69i 2.89-1.34i 2.36-4.25i 4.15+0.00i -0.02-0.46i -1.89+1.15i 0.04-3.69i -0.02+0.46i 0.33+0.00i ,  
A = 1.7-2.3i -1.8+2.4i 2.9-2.1i 1.2+1.4i -2.9+1.0i 0.6+0.8i 1.5+0.9i -1.4-1.7i -0.3-1.9i 2.1-1.1i  
and
B = -2.4+1.4i 0.6-2.9i -0.2-2.9i -1.5+0.1i 3.5+0.8i 2.2+3.7i .  

10.1
Program Text

Program Text (f16zrce.c)

10.2
Program Data

Program Data (f16zrce.d)

10.3
Program Results

Program Results (f16zrce.r)

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