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<HTML>
<HEAD><TITLE>MB04GD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB04GD">MB04GD</A></H2>
<H3>
RQ factorization with row pivoting of a matrix
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
To compute an RQ factorization with row pivoting of a
real m-by-n matrix A: P*A = R*Q.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
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( * )
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
</PRE>
<B>Input/Output Parameters</B>
<PRE>
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.
</PRE>
<B>Workspace</B>
<PRE>
DWORK DOUBLE PRECISION array, dimension (3*M)
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal
value.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
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.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] 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.
SIAM, Philadelphia, 1995.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The algorithm is backward stable.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
* MB04GD EXAMPLE PROGRAM TEXT
*
* .. 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
</PRE>
<B>Program Data</B>
<PRE>
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
</PRE>
<B>Program Results</B>
<PRE>
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
</PRE>
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