## MB04UD

### Column echelon form by unitary transformations for a rectangular matrix (added functionality)

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

Purpose

```  To compute orthogonal transformations Q and Z such that the
transformed pencil Q'(sE-A)Z has the E matrix in column echelon
form, where E and A are M-by-N matrices.

```
Specification
```      SUBROUTINE MB04UD( JOBQ, JOBZ, M, N, A, LDA, E, LDE, Q, LDQ,
\$                   Z, LDZ, RANKE, ISTAIR, TOL, DWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER         JOBQ, JOBZ
INTEGER           INFO, LDA, LDE, LDQ, LDZ, M, N, RANKE
DOUBLE PRECISION  TOL
C     .. Array Arguments ..
INTEGER           ISTAIR(*)
DOUBLE PRECISION  A(LDA,*), DWORK(*), E(LDE,*), Q(LDQ,*), Z(LDZ,*)

```
Arguments

Mode Parameters

```  JOBQ    CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix Q the unitary row permutations, as follows:
= 'N':  Do not form Q;
= 'I':  Q is initialized to the unit matrix and the
unitary row permutation matrix Q is returned;
= 'U':  The given matrix Q is updated by the unitary
row permutations used in the reduction.

JOBZ    CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix Z the unitary column transformations, as follows:
= 'N':  Do not form Z;
= 'I':  Z is initialized to the unit matrix and the
unitary transformation matrix Z is returned;
= 'U':  The given matrix Z is updated by the unitary
transformations used in the reduction.

```
Input/Output Parameters
```  M       (input) INTEGER
The number of rows in the matrices A, E and the order of
the matrix Q.  M >= 0.

N       (input) INTEGER
The number of columns in the matrices A, E and the order
of the matrix Z.  N >= 0.

A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading M-by-N part of this array must
contain the A matrix of the pencil sE-A.
On exit, the leading M-by-N part of this array contains
the unitary transformed matrix Q' * A * Z.

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

E       (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading M-by-N part of this array must
contain the E matrix of the pencil sE-A, to be reduced to
column echelon form.
On exit, the leading M-by-N part of this array contains
the unitary transformed matrix Q' * E * Z, which is in
column echelon form.

LDE     INTEGER
The leading dimension of array E.  LDE >= MAX(1,M).

Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,*)
On entry, if JOBQ = 'U', then the leading M-by-M part of
this array must contain a given matrix Q (e.g. from a
previous call to another SLICOT routine), and on exit, the
leading M-by-M part of this array contains the product of
the input matrix Q and the row permutation matrix used to
transform the rows of matrix E.
On exit, if JOBQ = 'I', then the leading M-by-M part of
this array contains the matrix of accumulated unitary
row transformations performed.
If JOBQ = 'N', the array Q is not referenced and can be
supplied as a dummy array (i.e. set parameter LDQ = 1 and
declare this array to be Q(1,1) in the calling program).

LDQ     INTEGER
The leading dimension of array Q. If JOBQ = 'U' or
JOBQ = 'I', LDQ >= MAX(1,M); if JOBQ = 'N', LDQ >= 1.

Z       (input/output) DOUBLE PRECISION array, dimension (LDZ,*)
On entry, if JOBZ = 'U', then the leading N-by-N part of
this array must contain a given matrix Z (e.g. from a
previous call to another SLICOT routine), and on exit, the
leading N-by-N part of this array contains the product of
the input matrix Z and the column transformation matrix
used to transform the columns of matrix E.
On exit, if JOBZ = 'I', then the leading N-by-N part of
this array contains the matrix of accumulated unitary
column transformations performed.
If JOBZ = 'N', the array Z is not referenced and can be
supplied as a dummy array (i.e. set parameter LDZ = 1 and
declare this array to be Z(1,1) in the calling program).

LDZ     INTEGER
The leading dimension of array Z. If JOBZ = 'U' or
JOBZ = 'I', LDZ >= MAX(1,N); if JOBZ = 'N', LDZ >= 1.

RANKE   (output) INTEGER
The computed rank of the unitary transformed matrix E.

ISTAIR  (output) INTEGER array, dimension (M)
This array contains information on the column echelon form
of the unitary transformed matrix E. Specifically,
ISTAIR(i) = +j if the first non-zero element E(i,j)
is a corner point and -j otherwise, for i = 1,2,...,M.

```
Tolerances
```  TOL     DOUBLE PRECISION
A tolerance below which matrix elements are considered
to be zero. If the user sets TOL to be less than (or
equal to) zero then the tolerance is taken as
EPS * MAX(ABS(E(I,J))), where EPS is the machine
precision (see LAPACK Library routine DLAMCH),
I = 1,2,...,M and J = 1,2,...,N.

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (MAX(M,N))

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

```
Method
```  Given an M-by-N matrix pencil sE-A with E not necessarily regular,
the routine computes a unitary transformed pencil Q'(sE-A)Z such
that the matrix Q' * E * Z is in column echelon form (trapezoidal
form).  Further details can be found in .

[An M-by-N matrix E with rank(E) = r is said to be in column
echelon form if the following conditions are satisfied:
(a) the first (N - r) columns contain only zero elements; and
(b) if E(i(k),k) is the last nonzero element in column k for
k = N-r+1,...,N, i.e. E(i(k),k) <> 0 and E(j,k) = 0 for
j > i(k), then 1 <= i(N-r+1) < i(N-r+2) < ... < i(N) <= M.]

```
References
```   Beelen, Th. and Van Dooren, P.
An improved algorithm for the computation of Kronecker's
canonical form of a singular pencil.
Linear Algebra and Applications, 105, pp. 9-65, 1988.

```
Numerical Aspects
```  It is shown in  that the algorithm is numerically backward
stable. The operations count is proportional to (MAX(M,N))**3.

```
```  None
```
Example

Program Text

```*     MB04UD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
INTEGER          MMAX, NMAX
PARAMETER        ( MMAX = 20, NMAX = 20 )
INTEGER          LDA, LDE, LDQ, LDZ
PARAMETER        ( LDA = MMAX, LDE = MMAX, LDQ = MMAX,
\$                   LDZ = NMAX )
INTEGER          LDWORK
PARAMETER        ( LDWORK = MAX( NMAX,MMAX ) )
*     PARAMETER        ( LDWORK = NMAX+MMAX )
*     .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER          I, INFO, J, M, N, RANKE
CHARACTER*1      JOBQ, JOBZ
*     .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), E(LDE,NMAX),
\$                 Q(LDQ,MMAX), Z(LDZ,NMAX)
INTEGER          ISTAIR(MMAX)
*     .. External Subroutines ..
EXTERNAL         MB04UD
*     .. Intrinsic Functions ..
INTRINSIC        MAX
*     .. 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, TOL
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) M
ELSE IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,M )
READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,M )
JOBQ = 'N'
JOBZ = 'N'
*        Reduce E to column echelon form and compute Q'*A*Z.
CALL MB04UD( JOBQ, JOBZ, M, N, A, LDA, E, LDE, Q, LDQ, Z, LDZ,
\$                RANKE, ISTAIR, TOL, DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99991 )
DO 10 I = 1, M
WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,N )
10       CONTINUE
WRITE ( NOUT, FMT = 99997 )
DO 100 I = 1, M
WRITE ( NOUT, FMT = 99996 ) ( E(I,J), J = 1,N )
100       CONTINUE
WRITE ( NOUT, FMT = 99995 ) RANKE
WRITE ( NOUT, FMT = 99994 ) ( ISTAIR(I), I = 1,M )
END IF
END IF
STOP
*
99999 FORMAT (' MB04UD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB04UD = ',I2)
99997 FORMAT (' The transformed matrix E is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The computed rank of E = ',I2)
99994 FORMAT (/' ISTAIR is ',/20(1X,I5))
99993 FORMAT (/' M is out of range.',/' M = ',I5)
99992 FORMAT (/' N is out of range.',/' N = ',I5)
99991 FORMAT (' The transformed matrix A is ')
END
```
Program Data
``` MB04UD EXAMPLE PROGRAM DATA
4     4     0.0
2.0  0.0  2.0 -2.0
0.0 -2.0  0.0  2.0
2.0  0.0 -2.0  0.0
2.0 -2.0  0.0  2.0
1.0  0.0  1.0 -1.0
0.0 -1.0  0.0  1.0
1.0  0.0 -1.0  0.0
1.0 -1.0  0.0  1.0
```
Program Results
``` MB04UD EXAMPLE PROGRAM RESULTS

The transformed matrix A is
0.5164   1.0328   1.1547  -2.3094
0.0000  -2.5820   0.0000  -1.1547
0.0000   0.0000  -3.4641   0.0000
0.0000   0.0000   0.0000  -3.4641
The transformed matrix E is
0.2582   0.5164   0.5774  -1.1547
0.0000  -1.2910   0.0000  -0.5774
0.0000   0.0000  -1.7321   0.0000
0.0000   0.0000   0.0000  -1.7321

The computed rank of E =  4

ISTAIR is
1     2     3     4
```