NAG Library Routine Document

f08fgf (dormtr)


f08fgf (dormtr) multiplies an arbitrary real matrix C by the real orthogonal matrix Q which was determined by f08fef (dsytrd) when reducing a real symmetric matrix to tridiagonal form.


Fortran Interface
Subroutine f08fgf ( side, uplo, trans, m, n, a, lda, tau, c, ldc, work, lwork, info)
Integer, Intent (In):: m, n, lda, ldc, lwork
Integer, Intent (Out):: info
Real (Kind=nag_wp), Intent (In):: tau(*)
Real (Kind=nag_wp), Intent (Inout):: a(lda,*), c(ldc,*)
Real (Kind=nag_wp), Intent (Out):: work(max(1,lwork))
Character (1), Intent (In):: side, uplo, trans
C Header Interface
#include <nagmk26.h>
void  f08fgf_ (const char *side, const char *uplo, const char *trans, const Integer *m, const Integer *n, double a[], const Integer *lda, const double tau[], double c[], const Integer *ldc, double work[], const Integer *lwork, Integer *info, const Charlen length_side, const Charlen length_uplo, const Charlen length_trans)
The routine may be called by its LAPACK name dormtr.


f08fgf (dormtr) is intended to be used after a call to f08fef (dsytrd), which reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: A=QTQT. f08fef (dsytrd) represents the orthogonal matrix Q as a product of elementary reflectors.
This routine may be used to form one of the matrix products
QC , QTC , CQ ​ or ​ CQT ,  
overwriting the result on C (which may be any real rectangular matrix).
A common application of this routine 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:     side – Character(1)Input
On entry: indicates how Q or QT is to be applied to C.
Q or QT is applied to C from the left.
Q or QT is applied to C from the right.
Constraint: side='L' or 'R'.
2:     uplo – Character(1)Input
On entry: this must be the same argument uplo as supplied to f08fef (dsytrd).
Constraint: uplo='U' or 'L'.
3:     trans – Character(1)Input
On entry: indicates whether Q or QT is to be applied to C.
Q is applied to C.
QT is applied to C.
Constraint: trans='N' or 'T'.
4:     m – IntegerInput
On entry: m, the number of rows of the matrix C; m is also the order of Q if side='L'.
Constraint: m0.
5:     n – IntegerInput
On entry: n, the number of columns of the matrix C; n is also the order of Q if side='R'.
Constraint: n0.
6:     alda* – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array a must be at least max1,m if side='L' and at least max1,n if side='R'.
On entry: details of the vectors which define the elementary reflectors, as returned by f08fef (dsytrd).
7:     lda – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08fgf (dormtr) is called.
  • if side='L', lda max1,m ;
  • if side='R', lda max1,n .
8:     tau* – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array tau must be at least max1,m-1 if side='L' and at least max1,n-1 if side='R'.
On entry: further details of the elementary reflectors, as returned by f08fef (dsytrd).
9:     cldc* – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array c must be at least max1,n.
On entry: the m by n matrix C.
On exit: c is overwritten by QC or QTC or CQ or CQT as specified by side and trans.
10:   ldc – IntegerInput
On entry: the first dimension of the array c as declared in the (sub)program from which f08fgf (dormtr) is called.
Constraint: ldcmax1,m.
11:   workmax1,lwork – Real (Kind=nag_wp) arrayWorkspace
On exit: if info=0, work1 contains the minimum value of lwork required for optimal performance.
12:   lwork – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08fgf (dormtr) is called.
If lwork=-1, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
Suggested value: for optimal performance, lworkn×nb if side='L' and at least m×nb if side='R', where nb is the optimal block size.
  • if side='L', lworkmax1,n or lwork=-1;
  • if side='R', lworkmax1,m or lwork=-1.
13:   info – IntegerOutput
On exit: info=0 unless the routine detects an error (see Section 6).

Error Indicators and Warnings

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.


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

f08fgf (dormtr) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08fgf (dormtr) 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

Further Comments

The total number of floating-point operations is approximately 2m2n if side='L' and 2mn2 if side='R'.
The complex analogue of this routine is f08fuf (zunmtr).


This example computes the two smallest eigenvalues, and the associated eigenvectors, of the matrix A, where
A = 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81 .  
Here A is symmetric and must first be reduced to tridiagonal form T by f08fef (dsytrd). The program then calls f08jjf (dstebz) to compute the requested eigenvalues and f08jkf (dstein) to compute the associated eigenvectors of T. Finally f08fgf (dormtr) is called to transform the eigenvectors to those of A.

Program Text

Program Text (f08fgfe.f90)

Program Data

Program Data (f08fgfe.d)

Program Results

Program Results (f08fgfe.r)