NAG Library Function Document

nag_zupmtr (f08guc)


    1  Purpose
    7  Accuracy


nag_zupmtr (f08guc) multiplies an arbitrary complex matrix C by the complex unitary matrix Q which was determined by nag_zhptrd (f08gsc) when reducing a complex Hermitian matrix to tridiagonal form.


#include <nag.h>
#include <nagf08.h>
void  nag_zupmtr (Nag_OrderType order, Nag_SideType side, Nag_UploType uplo, Nag_TransType trans, Integer m, Integer n, Complex ap[], const Complex tau[], Complex c[], Integer pdc, NagError *fail)


nag_zupmtr (f08guc) is intended to be used after a call to nag_zhptrd (f08gsc), which reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: A=QTQH. nag_zhptrd (f08gsc) represents the unitary matrix Q as a product of elementary reflectors.
This function may be used to form one of the matrix products
QC , QHC , CQ ​ or ​ CQH ,  
overwriting the result on C (which may be any complex rectangular matrix).
A common application of this function is to transform a matrix Z of eigenvectors of T to the matrix QZ of eigenvectors of A.


Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore


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 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:     side Nag_SideTypeInput
On entry: indicates how Q or QH is to be applied to C.
Q or QH is applied to C from the left.
Q or QH is applied to C from the right.
Constraint: side=Nag_LeftSide or Nag_RightSide.
3:     uplo Nag_UploTypeInput
On entry: this must be the same argument uplo as supplied to nag_zhptrd (f08gsc).
Constraint: uplo=Nag_Upper or Nag_Lower.
4:     trans Nag_TransTypeInput
On entry: indicates whether Q or QH is to be applied to C.
Q is applied to C.
QH is applied to C.
Constraint: trans=Nag_NoTrans or Nag_ConjTrans.
5:     m IntegerInput
On entry: m, the number of rows of the matrix C; m is also the order of Q if side=Nag_LeftSide.
Constraint: m0.
6:     n IntegerInput
On entry: n, the number of columns of the matrix C; n is also the order of Q if side=Nag_RightSide.
Constraint: n0.
7:     ap[dim] ComplexInput/Output
Note: the dimension, dim, of the array ap must be at least
  • max1, m × m+1 / 2  when side=Nag_LeftSide;
  • max1, n × n+1 / 2  when side=Nag_RightSide.
On entry: details of the vectors which define the elementary reflectors, as returned by nag_zhptrd (f08gsc).
On exit: is used as internal workspace prior to being restored and hence is unchanged.
8:     tau[dim] const ComplexInput
Note: the dimension, dim, of the array tau must be at least
  • max1,m-1 when side=Nag_LeftSide;
  • max1,n-1 when side=Nag_RightSide.
On entry: further details of the elementary reflectors, as returned by nag_zhptrd (f08gsc).
9:     c[dim] ComplexInput/Output
Note: the dimension, dim, of the array c must be at least
  • max1,pdc×n when order=Nag_ColMajor;
  • max1,m×pdc when order=Nag_RowMajor.
The i,jth element of the matrix C is stored in
  • c[j-1×pdc+i-1] when order=Nag_ColMajor;
  • c[i-1×pdc+j-1] when order=Nag_RowMajor.
On entry: the m by n matrix C.
On exit: c is overwritten by QC or QHC or CQ or CQH as specified by side and trans.
10:   pdc IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
  • if order=Nag_ColMajor, pdcmax1,m;
  • if order=Nag_RowMajor, pdcmax1,n.
11:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, argument value had an illegal value.
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On entry, pdc=value.
Constraint: pdc>0.
On entry, pdc=value and m=value.
Constraint: pdcmax1,m.
On entry, pdc=value and n=value.
Constraint: pdcmax1,n.
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.
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.


The computed result differs from the exact result by a matrix E such that
E2 = Oε C2 ,  
where ε is the machine precision.

Parallelism and Performance

nag_zupmtr (f08guc) 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.

Further Comments

The total number of real floating-point operations is approximately 8m2n if side=Nag_LeftSide and 8mn2 if side=Nag_RightSide.
The real analogue of this function is nag_dopmtr (f08ggc).


This example computes the two smallest eigenvalues, and the associated eigenvectors, of the matrix A, where
A = -2.28+0.00i 1.78-2.03i 2.26+0.10i -0.12+2.53i 1.78+2.03i -1.12+0.00i 0.01+0.43i -1.07+0.86i 2.26-0.10i 0.01-0.43i -0.37+0.00i 2.31-0.92i -0.12-2.53i -1.07-0.86i 2.31+0.92i -0.73+0.00i ,  
using packed storage. Here A is Hermitian and must first be reduced to tridiagonal form T by nag_zhptrd (f08gsc). The program then calls nag_dstebz (f08jjc) to compute the requested eigenvalues and nag_zstein (f08jxc) to compute the associated eigenvectors of T. Finally nag_zupmtr (f08guc) is called to transform the eigenvectors to those of A.

Program Text

Program Text (f08guce.c)

Program Data

Program Data (f08guce.d)

Program Results

Program Results (f08guce.r)

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