NAG CL Interface
f16zrc (zher2k)

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1 Purpose

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

2 Specification

#include <nag.h>
void  f16zrc (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)
The function may be called by the names: f16zrc, nag_blast_zher2k or nag_zher2k.

3 Description

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×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 https://www.netlib.org/blas/blast-forum/blas-report.pdf

5 Arguments

1: order Nag_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: uplo Nag_UploType Input
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_TransType Input
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 Integer Input
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 Integer Input
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 Complex Input
On entry: the scalar α.
7: a[dim] const Complex Input
Note: the dimension, dim, of the array a must be at least
  • max(1,pda×k) when trans=Nag_NoTrans and order=Nag_ColMajor;
  • max(1,n×pda) when trans=Nag_NoTrans and order=Nag_RowMajor;
  • max(1,pda×n) when trans=Nag_ConjTrans and order=Nag_ColMajor;
  • max(1,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×k if trans=Nag_NoTrans, or k×n otherwise.
8: pda Integer Input
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 max(1,n) ;
    • if trans=Nag_ConjTrans, pda max(1,k) ;
  • if order=Nag_RowMajor,
    • if trans=Nag_NoTrans, pdamax(1,k);
    • if trans=Nag_ConjTrans, pdamax(1,n).
9: b[dim] const Complex Input
Note: the dimension, dim, of the array b must be at least
  • max(1,pdb×k) when trans=Nag_NoTrans and order=Nag_ColMajor;
  • max(1,n×pdb) when trans=Nag_NoTrans and order=Nag_RowMajor;
  • max(1,pdb×n) when trans=Nag_ConjTrans and order=Nag_ColMajor;
  • max(1,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×k if trans=Nag_NoTrans, or k×n otherwise.
10: pdb Integer Input
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 max(1,n) ;
    • if trans=Nag_ConjTrans, pdb max(1,k) ;
  • if order=Nag_RowMajor,
    • if trans=Nag_NoTrans, pdbmax(1,k);
    • if trans=Nag_ConjTrans, pdbmax(1,n).
11: beta double Input
On entry: the scalar β.
12: c[dim] Complex Input/Output
Note: the dimension, dim, of the array c must be at least max(1,pdc×n).
On entry: the n×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 Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix C in the array c.
Constraint: pdcmax(1,n).
14: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error 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 value had an illegal value.
NE_ENUM_INT_2
On entry, trans=value, k=value, pda=value.
Constraint: if trans=Nag_ConjTrans, pda max(1,k) .
On entry, trans=value, k=value, pda=value.
Constraint: if trans=Nag_NoTrans, pdamax(1,k).
On entry, trans=value, k=value, pdb=value.
Constraint: if trans=Nag_ConjTrans, pdb max(1,k) .
On entry, trans=value, k=value, pdb=value.
Constraint: if trans=Nag_NoTrans, pdbmax(1,k).
On entry, trans=value, n=value, pda=value.
Constraint: if trans=Nag_ConjTrans, pdamax(1,n).
On entry, trans=value, n=value, pda=value.
Constraint: if trans=Nag_NoTrans, pda max(1,n) .
On entry, trans=value, n=value, pdb=value.
Constraint: if trans=Nag_ConjTrans, pdbmax(1,n).
On entry, trans=value, n=value, pdb=value.
Constraint: if trans=Nag_NoTrans, pdb max(1,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: pdcmax(1,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 8 in the Introduction to the NAG Library CL Interface 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

Background information to multithreading can be found in the Multithreading documentation.
f16zrc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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 4×4 matrix C using 4×2 matrices A and B, C=-C+(-0.5+0.5i)ABT+(-0.5-0.5i)BAT, 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)