## MB3OYZ

### Matrix rank determination by incremental condition estimation, during the pivoted QR factorization process (complex case)

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

Purpose

```  To compute a rank-revealing QR factorization of a complex general
M-by-N matrix  A,  which may be rank-deficient, and estimate its
effective rank using incremental condition estimation.

The routine uses a truncated QR factorization with column pivoting
[ R11 R12 ]
A * P = Q * R,  where  R = [         ],
[  0  R22 ]
with R11 defined as the largest leading upper triangular submatrix
whose estimated condition number is less than 1/RCOND.  The order
of R11, RANK, is the effective rank of A.  Condition estimation is
performed during the QR factorization process.  Matrix R22 is full
(but of small norm), or empty.

MB3OYZ  does not perform any scaling of the matrix A.

```
Specification
```      SUBROUTINE MB3OYZ( M, N, A, LDA, RCOND, SVLMAX, RANK, SVAL, JPVT,
\$                   TAU, DWORK, ZWORK, INFO )
C     .. Scalar Arguments ..
INTEGER            INFO, LDA, M, N, RANK
DOUBLE PRECISION   RCOND, SVLMAX
C     .. Array Arguments ..
INTEGER            JPVT( * )
COMPLEX*16         A( LDA, * ), TAU( * ), ZWORK( * )
DOUBLE PRECISION   DWORK( * ), SVAL( 3 )

```
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) COMPLEX*16 array, dimension ( LDA, N )
On entry, the leading M-by-N part of this array must
contain the given matrix A.
On exit, the leading RANK-by-RANK upper triangular part
of A contains the triangular factor R11, and the elements
below the diagonal in the first  RANK  columns, with the
array TAU, represent the unitary matrix Q as a product
of  RANK  elementary reflectors.
The remaining  N-RANK  columns contain the result of the
QR factorization process used.

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

RCOND   (input) DOUBLE PRECISION
RCOND is used to determine the effective rank of A, which
is defined as the order of the largest leading triangular
submatrix R11 in the QR factorization with pivoting of A,
whose estimated condition number is less than 1/RCOND.
0 <= RCOND <= 1.
NOTE that when SVLMAX > 0, the estimated rank could be
less than that defined above (see SVLMAX).

SVLMAX  (input) DOUBLE PRECISION
If A is a submatrix of another matrix B, and the rank
decision should be related to that matrix, then SVLMAX
should be an estimate of the largest singular value of B
(for instance, the Frobenius norm of B).  If this is not
the case, the input value SVLMAX = 0 should work.
SVLMAX >= 0.

RANK    (output) INTEGER
The effective (estimated) rank of A, i.e., the order of
the submatrix R11.

SVAL    (output) DOUBLE PRECISION array, dimension ( 3 )
The estimates of some of the singular values of the
triangular factor R:
SVAL(1): largest singular value of R(1:RANK,1:RANK);
SVAL(2): smallest singular value of R(1:RANK,1:RANK);
SVAL(3): smallest singular value of R(1:RANK+1,1:RANK+1),
if RANK < MIN( M, N ), or of R(1:RANK,1:RANK),
otherwise.
If the triangular factorization is a rank-revealing one
(which will be the case if the leading columns were well-
conditioned), then SVAL(1) will also be an estimate for
the largest singular value of A, and SVAL(2) and SVAL(3)
will be estimates for the RANK-th and (RANK+1)-st singular
values of A, respectively.
By examining these values, one can confirm that the rank
is well defined with respect to the chosen value of RCOND.
The ratio SVAL(1)/SVAL(2) is an estimate of the condition
number of R(1:RANK,1:RANK).

JPVT    (output) INTEGER array, dimension ( N )
If JPVT(i) = k, then the i-th column of A*P was the k-th
column of A.

TAU     (output) COMPLEX*16 array, dimension ( MIN( M, N ) )
The leading  RANK  elements of TAU contain the scalar
factors of the elementary reflectors.

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension ( 2*N )

ZWORK   COMPLEX*16 array, dimension ( 3*N-1 )

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

```
Method
```  The routine computes a truncated QR factorization with column
pivoting of A,  A * P = Q * R,  with  R  defined above, and,
during this process, finds the largest leading submatrix whose
estimated condition number is less than 1/RCOND, taking the
possible positive value of SVLMAX into account.  This is performed
using the LAPACK incremental condition estimation scheme and a
slightly modified rank decision test.  The factorization process
stops when  RANK  has been determined.

The matrix Q is represented as a product of elementary reflectors

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

Each H(i) has the form

H = I - tau * v * v'

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

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

```
References
```  [1] Bischof, C.H. and P. Tang.
Generalizing Incremental Condition Estimation.
LAPACK Working Notes 32, Mathematics and Computer Science
Division, Argonne National Laboratory, UT, CS-91-132,
May 1991.

[2] Bischof, C.H. and P. Tang.
Robust Incremental Condition Estimation.
LAPACK Working Notes 33, Mathematics and Computer Science
Division, Argonne National Laboratory, UT, CS-91-133,
May 1991.

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

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```  None
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Example

Program Text

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Program Data
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Program Results
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