## MB02JX

### Low rank QR factorization with column pivoting of a block Toeplitz matrix

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

Purpose

```  To compute a low rank QR factorization with column pivoting of a
K*M-by-L*N block Toeplitz matrix T with blocks of size (K,L);
specifically,
T
T P =  Q R ,

where R is lower trapezoidal, P is a block permutation matrix
and Q^T Q = I. The number of columns in R is equivalent to the
numerical rank of T with respect to the given tolerance TOL1.
Note that the pivoting scheme is local, i.e., only columns
belonging to the same block in T are permuted.

```
Specification
```      SUBROUTINE MB02JX( JOB, K, L, M, N, TC, LDTC, TR, LDTR, RNK, Q,
\$                   LDQ, R, LDR, JPVT, TOL1, TOL2, DWORK, LDWORK,
\$                   INFO )
C     .. Scalar Arguments ..
CHARACTER         JOB
INTEGER           INFO, K, L, LDQ, LDR, LDTC, LDTR, LDWORK, M, N,
\$                  RNK
DOUBLE PRECISION  TOL1, TOL2
C     .. Array Arguments ..
DOUBLE PRECISION  DWORK(LDWORK), Q(LDQ,*), R(LDR,*), TC(LDTC,*),
\$                  TR(LDTR,*)
INTEGER           JPVT(*)

```
Arguments

Mode Parameters

```  JOB     CHARACTER*1
Specifies the output of the routine as follows:
= 'Q':  computes Q and R;
= 'R':  only computes R.

```
Input/Output Parameters
```  K       (input)  INTEGER
The number of rows in one block of T.  K >= 0.

L       (input)  INTEGER
The number of columns in one block of T.  L >= 0.

M       (input)  INTEGER
The number of blocks in one block column of T.  M >= 0.

N       (input)  INTEGER
The number of blocks in one block row of T.  N >= 0.

TC      (input) DOUBLE PRECISION array, dimension (LDTC, L)
The leading M*K-by-L part of this array must contain
the first block column of T.

LDTC    INTEGER
The leading dimension of the array TC.
LDTC >= MAX(1,M*K).

TR      (input)  DOUBLE PRECISION array, dimension (LDTR,(N-1)*L)
The leading K-by-(N-1)*L part of this array must contain
the first block row of T without the leading K-by-L
block.

LDTR    INTEGER
The leading dimension of the array TR.  LDTR >= MAX(1,K).

RNK     (output)  INTEGER
The number of columns in R, which is equivalent to the
numerical rank of T.

Q       (output)  DOUBLE PRECISION array, dimension (LDQ,RNK)
If JOB = 'Q', then the leading M*K-by-RNK part of this
array contains the factor Q.
If JOB = 'R', then this array is not referenced.

LDQ     INTEGER
The leading dimension of the array Q.
LDQ >= MAX(1,M*K),  if JOB = 'Q';
LDQ >= 1,           if JOB = 'R'.

R       (output)  DOUBLE PRECISION array, dimension (LDR,RNK)
The leading N*L-by-RNK part of this array contains the
lower trapezoidal factor R.

LDR     INTEGER
The leading dimension of the array R.
LDR >= MAX(1,N*L)

JPVT    (output)  INTEGER array, dimension (MIN(M*K,N*L))
This array records the column pivoting performed.
If JPVT(j) = k, then the j-th column of T*P was
the k-th column of T.

```
Tolerances
```  TOL1    DOUBLE PRECISION
If TOL1 >= 0.0, the user supplied diagonal tolerance;
if TOL1 < 0.0, a default diagonal tolerance is used.

TOL2    DOUBLE PRECISION
If TOL2 >= 0.0, the user supplied offdiagonal tolerance;
if TOL2 < 0.0, a default offdiagonal tolerance is used.

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK;  DWORK(2) and DWORK(3) return the used values
for TOL1 and TOL2, respectively.
On exit, if INFO = -19,  DWORK(1) returns the minimum
value of LDWORK.

LDWORK  INTEGER
The length of the array DWORK.
LDWORK >= MAX( 3, ( M*K + ( N - 1 )*L )*( L + 2*K ) + 9*L
+ MAX(M*K,(N-1)*L) ),    if JOB = 'Q';
LDWORK >= MAX( 3, ( N - 1 )*L*( L + 2*K + 1 ) + 9*L,
M*K*( L + 1 ) + L ),     if JOB = 'R'.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value;
= 1:  due to perturbations induced by roundoff errors, or
removal of nearly linearly dependent columns of the
generator, the Schur algorithm encountered a
situation where a diagonal element in the negative
generator is larger in magnitude than the
corresponding diagonal element in the positive
generator (modulo TOL1);
= 2:  due to perturbations induced by roundoff errors, or
removal of nearly linearly dependent columns of the
generator, the Schur algorithm encountered a
situation where diagonal elements in the positive
and negative generator are equal in magnitude
(modulo TOL1), but the offdiagonal elements suggest
that these columns are not linearly dependent
(modulo TOL2*ABS(diagonal element)).

```
Method
```  Householder transformations and modified hyperbolic rotations
are used in the Schur algorithm , .
If, during the process, the hyperbolic norm of a row in the
leading part of the generator is found to be less than or equal
to TOL1, then this row is not reduced. If the difference of the
corresponding columns has a norm less than or equal to TOL2 times
the magnitude of the leading element, then this column is removed
from the generator, as well as from R. Otherwise, the algorithm
breaks down. TOL1 is set to norm(TC)*sqrt(eps) and TOL2 is set
to N*L*sqrt(eps) by default.
If M*K > L, the columns of T are permuted so that the diagonal
elements in one block column of R have decreasing magnitudes.

```
References
```   Kailath, T. and Sayed, A.
Fast Reliable Algorithms for Matrices with Structure.

 Kressner, D. and Van Dooren, P.
Factorizations and linear system solvers for matrices with
Toeplitz structure.
SLICOT Working Note 2000-2, 2000.

```
Numerical Aspects
```  The algorithm requires 0(K*RNK*L*M*N) floating point operations.

```
```  None
```
Example

Program Text

```*     MB02JX EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
INTEGER          KMAX, LMAX, MMAX, NMAX
PARAMETER        ( KMAX = 20, LMAX = 20, MMAX = 20, NMAX = 20 )
INTEGER          LDR, LDQ, LDTC, LDTR, LDWORK
PARAMETER        ( LDR  = NMAX*LMAX, LDQ  = MMAX*KMAX,
\$                   LDTC = MMAX*KMAX, LDTR = KMAX,
\$                   LDWORK = ( MMAX*KMAX + NMAX*LMAX )
\$                            *( LMAX + 2*KMAX ) + 5*LMAX
\$                            + MMAX*KMAX + NMAX*LMAX )
*     .. Local Scalars ..
CHARACTER        JOB
INTEGER          I, INFO, J, K, L, M, N, RNK
DOUBLE PRECISION TOL1, TOL2
*     .. Local Arrays ..
INTEGER          JPVT(NMAX*LMAX)
DOUBLE PRECISION DWORK(LDWORK), Q(LDQ,NMAX*LMAX),
\$                 R(LDR,NMAX*LMAX), TC(LDTC,LMAX),
\$                 TR(LDTR,NMAX*LMAX)
*     .. External Functions ..
LOGICAL          LSAME
EXTERNAL         LSAME
*     .. External Subroutines ..
EXTERNAL         MB02JX
*     .. 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 = * ) K, L, M, N, TOL1, TOL2, JOB
IF( K.LE.0 .OR. K.GT.KMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) K
ELSE IF( L.LE.0 .OR. L.GT.LMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) L
ELSE IF( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE IF( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) N
ELSE
READ ( NIN, FMT = * ) ( ( TC(I,J), J = 1,L ), I = 1,M*K )
READ ( NIN, FMT = * ) ( ( TR(I,J), J = 1,( N - 1 )*L ),
\$                             I = 1,K )
*        Compute the required part of the QR factorization.
CALL MB02JX( JOB, K, L, M, N, TC, LDTC, TR, LDTR, RNK, Q, LDQ,
\$                R, LDR, JPVT, TOL1, TOL2, DWORK, LDWORK, INFO )
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) RNK
IF ( LSAME( JOB, 'Q' ) ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 10  I = 1, M*K
WRITE ( NOUT, FMT = 99993 ) ( Q(I,J), J = 1, RNK )
10          CONTINUE
END IF
WRITE ( NOUT, FMT = 99996 )
DO 20  I = 1, N*L
WRITE ( NOUT, FMT = 99993 ) ( R(I,J), J = 1, RNK )
20       CONTINUE
WRITE ( NOUT, FMT = 99995 )
WRITE ( NOUT, FMT = 99992 ) ( JPVT(I),
\$                                    I = 1, MIN( M*K, N*L ) )
END IF
END IF
STOP
*
99999 FORMAT (' MB02JX EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB02JX = ',I2)
99997 FORMAT (/' The factor Q is ')
99996 FORMAT (/' The factor R is ')
99995 FORMAT (/' The column permutation is ')
99994 FORMAT (/' Numerical rank ',/' RNK = ',I5)
99993 FORMAT (20(1X,F8.4))
99992 FORMAT (20(1X,I4))
99991 FORMAT (/' K is out of range.',/' K = ',I5)
99990 FORMAT (/' L is out of range.',/' L = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' N is out of range.',/' N = ',I5)
END
```
Program Data
```MB02JX EXAMPLE PROGRAM DATA
3   3   4   4  -1.0D0  -1.0D0   Q
1.0     2.0     3.0
1.0     2.0     3.0
1.0     2.0     3.0
1.0     2.0     3.0
1.0     2.0     3.0
1.0     2.0     3.0
1.0     2.0     3.0
1.0     2.0     3.0
1.0     2.0     3.0
1.0     0.0     1.0
1.0     1.0     0.0
2.0     2.0     0.0
1.0     2.0     3.0     1.0     2.0     3.0     0.0     1.0     1.0
1.0     2.0     3.0     1.0     2.0     3.0     1.0     2.0     1.0
1.0     2.0     3.0     1.0     2.0     3.0     1.0     1.0     1.0
1.0     2.0     3.0     1.0     2.0     3.0     0.0     1.0     0.0
```
Program Results
``` MB02JX EXAMPLE PROGRAM RESULTS

Numerical rank
RNK =     7

The factor Q is
-0.3313  -0.0105  -0.0353   0.0000  -0.4714  -0.8165   0.0000
-0.3313  -0.0105  -0.0353   0.0000  -0.4714   0.4082   0.7071
-0.3313  -0.0105  -0.0353   0.0000  -0.4714   0.4082  -0.7071
-0.3313  -0.0105  -0.0353   0.0000   0.2357   0.0000   0.0000
-0.3313  -0.0105  -0.0353   0.0000   0.2357   0.0000   0.0000
-0.3313  -0.0105  -0.0353   0.0000   0.2357   0.0000   0.0000
-0.3313  -0.0105  -0.0353   0.0000   0.2357   0.0000   0.0000
-0.3313  -0.0105  -0.0353   0.0000   0.2357   0.0000   0.0000
-0.3313  -0.0105  -0.0353   0.0000   0.2357   0.0000   0.0000
-0.1104   0.2824   0.9529   0.0000   0.0000   0.0000   0.0000
0.0000   0.4288  -0.1271   0.8944   0.0000   0.0000   0.0000
0.0000   0.8576  -0.2541  -0.4472   0.0000   0.0000   0.0000

The factor R is
-9.0554   0.0000   0.0000   0.0000   0.0000   0.0000   0.0000
-3.0921   2.3322   0.0000   0.0000   0.0000   0.0000   0.0000
-5.9633   1.9557  -1.2706   0.0000   0.0000   0.0000   0.0000
-9.2762   4.4238   0.7623   1.3416   0.0000   0.0000   0.0000
-6.1842   2.9492   0.5082   0.8944   0.0000   0.0000   0.0000
-3.0921   1.4746   0.2541   0.4472   0.0000   0.0000   0.0000
-9.2762   4.4238   0.7623   1.3416   0.0000   0.0000   0.0000
-6.1842   2.9492   0.5082   0.8944   0.0000   0.0000   0.0000
-3.0921   1.4746   0.2541   0.4472   0.0000   0.0000   0.0000
-7.2885   4.4866   0.9741   1.3416   2.8284   0.0000   0.0000
-2.7608   1.4851   0.2894   0.4472   0.4714   0.8165   0.0000
-5.5216   2.9701   0.5788   0.8944   0.9428   0.4082   0.7071

The column permutation is
3    1    2    6    5    4    9    8    7   12   10   11
```