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<H2><A Name="MB02SD">MB02SD</A></H2>
<H3>
LU factorization of an upper Hessenberg 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 LU factorization of an n-by-n upper Hessenberg
matrix H using partial pivoting with row interchanges.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB02SD( N, H, LDH, IPIV, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDH, N
C .. Array Arguments ..
INTEGER IPIV(*)
DOUBLE PRECISION H(LDH,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the matrix H. N >= 0.
H (input/output) DOUBLE PRECISION array, dimension (LDH,N)
On entry, the n-by-n upper Hessenberg matrix to be
factored.
On exit, the factors L and U from the factorization
H = P*L*U; the unit diagonal elements of L are not stored,
and L is lower bidiagonal.
LDH INTEGER
The leading dimension of the array H. LDH >= max(1,N).
IPIV (output) INTEGER array, dimension (N)
The pivot indices; for 1 <= i <= N, row i of the matrix
was interchanged with row IPIV(i).
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, U(i,i) is exactly zero. The
factorization has been completed, but the factor U
is exactly singular, and division by zero will occur
if it is used to solve a system of equations.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The factorization has the form
H = P * L * U
where P is a permutation matrix, L is lower triangular with unit
diagonal elements (and one nonzero subdiagonal), and U is upper
triangular.
This is the right-looking Level 1 BLAS version of the algorithm
(adapted after DGETF2).
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
-
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE> 2
The algorithm requires 0( N ) operations.
</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>
* MB02SD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, NRHMAX
PARAMETER ( NMAX = 20, NRHMAX = 20 )
INTEGER LDB, LDH
PARAMETER ( LDB = NMAX, LDH = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = 3*NMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = NMAX )
* .. Local Scalars ..
DOUBLE PRECISION HNORM, RCOND
INTEGER I, INFO, INFO1, J, N, NRHS
CHARACTER*1 NORM, TRANS
* .. Local Arrays ..
DOUBLE PRECISION H(LDH,NMAX), B(LDB,NRHMAX), DWORK(LDWORK)
INTEGER IPIV(NMAX), IWORK(LIWORK)
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLANHS
EXTERNAL DLAMCH, DLANHS
* .. External Subroutines ..
EXTERNAL DLASET, MB02RD, MB02SD, MB02TD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, NRHS, NORM, TRANS
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( H(I,J), J = 1,N ), I = 1,N )
IF ( NRHS.LT.0 .OR. NRHS.GT.NRHMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) NRHS
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,NRHS ), I = 1,N )
IF ( N.GT.2 )
$ CALL DLASET( 'Lower', N-2, N-2, ZERO, ZERO, H(3,1), LDH )
* Compute the LU factorization of the upper Hessenberg matrix.
CALL MB02SD( N, H, LDH, IPIV, INFO )
* Estimate the reciprocal condition number of the matrix.
HNORM = DLANHS( NORM, N, H, LDH, DWORK )
CALL MB02TD( NORM, N, HNORM, H, LDH, IPIV, RCOND, IWORK,
$ DWORK, INFO1 )
IF ( INFO.EQ.0 .AND. RCOND.GT.DLAMCH( 'Epsilon' ) ) THEN
* Solve the linear system.
CALL MB02RD( TRANS, N, NRHS, H, LDH, IPIV, B, LDB, INFO )
*
WRITE ( NOUT, FMT = 99997 )
ELSE
WRITE ( NOUT, FMT = 99998 ) INFO
END IF
DO 10 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( B(I,J), J = 1,NRHS )
10 CONTINUE
WRITE ( NOUT, FMT = 99995 ) RCOND
END IF
END IF
STOP
*
99999 FORMAT (' MB02SD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB02SD = ',I2)
99997 FORMAT (' The solution matrix is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' Reciprocal condition number = ',D12.4)
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' NRHS is out of range.',/' NRHS = ',I5)
END
</PRE>
<B>Program Data</B>
<PRE>
MB02SD EXAMPLE PROGRAM DATA
5 4 O N
1. 2. 6. 3. 5.
-2. -1. -1. 0. -2.
0. 3. 1. 5. 1.
0. 0. 2. 0. -4.
0. 0. 0. 1. 4.
5. 5. 1. 5.
-2. 1. 3. 1.
0. 0. 4. 5.
2. 1. 1. 3.
-1. 3. 3. 1.
</PRE>
<B>Program Results</B>
<PRE>
MB02SD EXAMPLE PROGRAM RESULTS
The solution matrix is
0.0435 1.2029 1.6377 1.1014
1.0870 -4.4275 -5.5580 -2.9638
0.9130 0.7609 -0.1087 0.6304
-0.8261 2.4783 4.2174 2.7391
-0.0435 0.1304 -0.3043 -0.4348
Reciprocal condition number = 0.1554D-01
</PRE>
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