## MB04GD

### RQ factorization with row pivoting of a matrix

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

```  To compute an RQ factorization with row pivoting of a
real m-by-n matrix A: P*A = R*Q.

```
Specification
```      SUBROUTINE MB04GD( M, N, A, LDA, JPVT, TAU, DWORK, INFO )
C     .. Scalar Arguments ..
INTEGER            INFO, LDA, M, N
C     .. Array Arguments ..
INTEGER            JPVT( * )
DOUBLE PRECISION   A( LDA, * ), DWORK( * ), TAU( * )

```
Arguments

Input/Output Parameters

```  M       (input) INTEGER
The number of rows of the matrix A.  M >= 0.

N       (input) INTEGER
The number of columns of the matrix A.  N >= 0.

A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the m-by-n matrix A.
On exit,
if m <= n, the upper triangle of the subarray
A(1:m,n-m+1:n) contains the m-by-m upper triangular
matrix R;
if m >= n, the elements on and above the (m-n)-th
subdiagonal contain the m-by-n upper trapezoidal matrix R;
the remaining elements, with the array TAU, represent the
orthogonal matrix Q as a product of min(m,n) elementary
reflectors (see METHOD).

LDA     INTEGER
The leading dimension of the array A. LDA >= max(1,M).

JPVT    (input/output) INTEGER array, dimension (M)
On entry, if JPVT(i) .ne. 0, the i-th row of A is permuted
to the bottom of P*A (a trailing row); if JPVT(i) = 0,
the i-th row of A is a free row.
On exit, if JPVT(i) = k, then the i-th row of P*A
was the k-th row of A.

TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors.

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (3*M)

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal
value.

```
Method
```  The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H = I - tau * v * v'

where tau is a real scalar, and v is a real vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit
in A(m-k+i,1:n-k+i-1), and tau in TAU(i).

The matrix P is represented in jpvt as follows: If
jpvt(j) = i
then the jth row of P is the ith canonical unit vector.

```
References
```   Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.

```
Numerical Aspects
```  The algorithm is backward stable.

```
```  None
```
Example

Program Text

```*     MB04GD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER        ( ZERO = 0.0D0 )
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
INTEGER          NMAX, MMAX
PARAMETER        ( NMAX = 10, MMAX = 10 )
INTEGER          LDA
PARAMETER        ( LDA = MMAX )
INTEGER          LDTAU
PARAMETER        ( LDTAU = MIN(MMAX,NMAX) )
INTEGER          LDWORK
PARAMETER        ( LDWORK = 3*MMAX )
*     .. Local Scalars ..
INTEGER          I, INFO, J, M, N
*     .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), TAU(LDTAU)
INTEGER          JPVT(MMAX)
*     .. External Subroutines ..
EXTERNAL         DLASET, MB04GD
*     .. Intrinsic Functions ..
INTRINSIC        MIN
*     .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, N
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99972 ) N
ELSE
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99971 ) M
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,M )
READ ( NIN, FMT = * ) ( JPVT(I), I = 1,M )
*           RQ with row pivoting.
CALL MB04GD( M, N, A, LDA, JPVT, TAU, DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) ( JPVT(I), I = 1,M )
WRITE ( NOUT, FMT = 99990 )
IF ( M.GE.N ) THEN
IF ( N.GT.1 )
\$               CALL DLASET( 'Lower', N-1, N-1, ZERO, ZERO,
\$                            A(M-N+2,1), LDA )
ELSE
CALL DLASET( 'Full', M, N-M-1, ZERO, ZERO, A, LDA )
CALL DLASET( 'Lower', M, M, ZERO, ZERO, A(1,N-M),
\$                          LDA )
END IF
DO 20 I = 1, M
WRITE ( NOUT, FMT = 99989 ) ( A(I,J), J = 1,N )
20          CONTINUE
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' MB04GD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB04GD = ',I2)
99994 FORMAT (' Row permutations are ',/(20(I3,2X)))
99990 FORMAT (/' The matrix A is ')
99989 FORMAT (20(1X,F8.4))
99972 FORMAT (/' N is out of range.',/' N = ',I5)
99971 FORMAT (/' M is out of range.',/' M = ',I5)
END
```
Program Data
``` MB04GD EXAMPLE PROGRAM DATA
6     5
1.    2.    6.    3.    5.
-2.   -1.   -1.    0.   -2.
5.    5.    1.    5.    1.
-2.   -1.   -1.    0.   -2.
4.    8.    4.   20.    4.
-2.   -1.   -1.    0.   -2.
0     0     0     0     0     0
```
Program Results
``` MB04GD EXAMPLE PROGRAM RESULTS

Row permutations are
2    4    6    3    1    5

The matrix A is
0.0000  -1.0517  -1.8646  -1.9712   1.2374
0.0000  -1.0517  -1.8646  -1.9712   1.2374
0.0000  -1.0517  -1.8646  -1.9712   1.2374
0.0000   0.0000   4.6768   0.0466  -7.4246
0.0000   0.0000   0.0000   6.7059  -5.4801
0.0000   0.0000   0.0000   0.0000 -22.6274
```