## NF01BQ

### Solving the linear system J x = b, D x = 0, D diagonal, for Wiener system identification

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

Purpose

```  To determine a vector x which solves the system of linear
equations

J*x = b ,     D*x = 0 ,

in the least squares sense, where J is an m-by-n matrix,
D is an n-by-n diagonal matrix, and b is an m-vector. The matrix J
is the current Jacobian of a nonlinear least squares problem,
provided in a compressed form by SLICOT Library routine NF01BD.
It is assumed that a block QR factorization, with column pivoting,
of J is available, that is, J*P = Q*R, where P is a permutation
matrix, Q has orthogonal columns, and R is an upper triangular
matrix with diagonal elements of nonincreasing magnitude for each
block, as returned by SLICOT Library routine NF01BS. The routine
NF01BQ needs the upper triangle of R in compressed form, the
permutation matrix P, and the first n components of Q'*b
(' denotes the transpose). The system J*x = b, D*x = 0, is then
equivalent to

R*z = Q'*b ,  P'*D*P*z = 0 ,                             (1)

where x = P*z. If this system does not have full rank, then an
approximate least squares solution is obtained (see METHOD).
On output, NF01BQ also provides an upper triangular matrix S
such that

P'*(J'*J + D*D)*P = S'*S .

The system (1) is equivalent to S*z = c , where c contains the
first n components of the vector obtained by applying to
[ (Q'*b)'  0 ]' the transformations which triangularized
[ R'  P'*D*P ]', getting S.

The matrix R has the following structure

/   R_1    0    ..   0   |   L_1   \
|    0    R_2   ..   0   |   L_2   |
|    :     :    ..   :   |    :    | ,
|    0     0    ..  R_l  |   L_l   |
\    0     0    ..   0   |  R_l+1  /

where the submatrices R_k, k = 1:l, have the same order BSN,
and R_k, k = 1:l+1, are square and upper triangular. This matrix
is stored in the compressed form

/   R_1  |   L_1   \
|   R_2  |   L_2   |
Rc =   |    :   |    :    | ,
|   R_l  |   L_l   |
\    X   |  R_l+1  /

where the submatrix X is irrelevant. The matrix S has the same
structure as R, and its diagonal blocks are denoted by S_k,
k = 1:l+1.

If l <= 1, then the full upper triangle of the matrix R is stored.

```
Specification
```      SUBROUTINE NF01BQ( COND, N, IPAR, LIPAR, R, LDR, IPVT, DIAG, QTB,
\$                   RANKS, X, TOL, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER         COND
INTEGER           INFO, LDR, LDWORK, LIPAR, N
DOUBLE PRECISION  TOL
C     .. Array Arguments ..
INTEGER           IPAR(*), IPVT(*), RANKS(*)
DOUBLE PRECISION  DIAG(*), DWORK(*), QTB(*), R(LDR,*), X(*)

```
Arguments

Mode Parameters

```  COND    CHARACTER*1
Specifies whether the condition of the matrices S_k should
be estimated, as follows:
= 'E' :  use incremental condition estimation and store
the numerical rank of S_k in the array entry
RANKS(k), for k = 1:l+1;
= 'N' :  do not use condition estimation, but check the
diagonal entries of S_k for zero values;
= 'U' :  use the ranks already stored in RANKS(1:l+1).

```
Input/Output Parameters
```  N       (input) INTEGER
The order of the matrix R.  N = BN*BSN + ST >= 0.
(See parameter description below.)

IPAR    (input) INTEGER array, dimension (LIPAR)
The integer parameters describing the structure of the
matrix R, as follows:
IPAR(1) must contain ST, the number of columns of the
submatrices L_k and the order of R_l+1.  ST >= 0.
IPAR(2) must contain BN, the number of blocks, l, in the
block diagonal part of R.  BN >= 0.
IPAR(3) must contain BSM, the number of rows of the blocks
R_k, k = 1:l.  BSM >= 0.
IPAR(4) must contain BSN, the number of columns of the
blocks R_k, k = 1:l.  BSN >= 0.
BSM is not used by this routine, but assumed equal to BSN.

LIPAR   (input) INTEGER
The length of the array IPAR.  LIPAR >= 4.

R       (input/output) DOUBLE PRECISION array, dimension (LDR, NC)
where NC = N if BN <= 1, and NC = BSN+ST, if BN > 1.
On entry, the leading N-by-NC part of this array must
contain the (compressed) representation (Rc) of the upper
triangular matrix R. If BN > 1, the submatrix X in Rc is
not referenced. The zero strict lower triangles of R_k,
k = 1:l+1, need not be set. If BN <= 1 or BSN = 0, then
the full upper triangle of R must be stored.
On exit, the full upper triangles of R_k, k = 1:l+1, and
L_k, k = 1:l, are unaltered, and the strict lower
triangles of R_k, k = 1:l+1, contain the corresponding
strict upper triangles (transposed) of the upper
triangular matrix S.
If BN <= 1 or BSN = 0, then the transpose of the strict
upper triangle of S is stored in the strict lower triangle
of R.

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

IPVT    (input) INTEGER array, dimension (N)
This array must define the permutation matrix P such that
J*P = Q*R. Column j of P is column IPVT(j) of the identity
matrix.

DIAG    (input) DOUBLE PRECISION array, dimension (N)
This array must contain the diagonal elements of the
matrix D.

QTB     (input) DOUBLE PRECISION array, dimension (N)
This array must contain the first n elements of the
vector Q'*b.

RANKS   (input or output) INTEGER array, dimension (r), where
r = BN + 1,  if ST > 0, BSN > 0, and BN > 1;
r = BN,      if ST = 0 and BSN > 0;
r = 1,       if ST > 0 and ( BSN = 0 or BN <= 1 );
r = 0,       if ST = 0 and BSN = 0.
On entry, if COND = 'U' and N > 0, this array must contain
the numerical ranks of the submatrices S_k, k = 1:l(+1).
On exit, if COND = 'E' or 'N' and N > 0, this array
contains the numerical ranks of the submatrices S_k,
k = 1:l(+1), estimated according to the value of COND.

X       (output) DOUBLE PRECISION array, dimension (N)
This array contains the least squares solution of the
system J*x = b, D*x = 0.

```
Tolerances
```  TOL     DOUBLE PRECISION
If COND = 'E', the tolerance to be used for finding the
ranks of the submatrices S_k. If the user sets TOL > 0,
then the given value of TOL is used as a lower bound for
the reciprocal condition number;  a (sub)matrix whose
estimated condition number is less than 1/TOL is
considered to be of full rank.  If the user sets TOL <= 0,
then an implicitly computed, default tolerance, defined by
TOLDEF = N*EPS,  is used instead, where EPS is the machine
precision (see LAPACK Library routine DLAMCH).
This parameter is not relevant if COND = 'U' or 'N'.

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, the first N elements of this array contain the
diagonal elements of the upper triangular matrix S, and
the next N elements contain the solution z.
If BN > 1 and BSN > 0, the elements 2*N+1 : 2*N+ST*(N-ST)
contain the submatrix (S(1:N-ST,N-ST+1:N))' of the
matrix S.

LDWORK  INTEGER
The length of the array DWORK.
LDWORK >= 2*N,              if BN <= 1 or  BSN = 0 and
COND <> 'E';
LDWORK >= 4*N,              if BN <= 1 or  BSN = 0 and
COND =  'E';
LDWORK >= ST*(N-ST) + 2*N,  if BN >  1 and BSN > 0 and
COND <> 'E';
LDWORK >= ST*(N-ST) + 2*N + 2*MAX(BSN,ST),
if BN >  1 and BSN > 0 and
COND = 'E'.

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

```
Method
```  Standard plane rotations are used to annihilate the elements of
the diagonal matrix D, updating the upper triangular matrix R
and the first n elements of the vector Q'*b. A basic least squares
solution is computed. The computations exploit the special
structure and storage scheme of the matrix R. If one or more of
the submatrices S_k, k = 1:l+1, is singular, then the computed
result is not the basic least squares solution for the whole
problem, but a concatenation of (least squares) solutions of the
individual subproblems involving R_k, k = 1:l+1 (with adapted
right hand sides).

```
References
```   More, J.J., Garbow, B.S, and Hillstrom, K.E.
User's Guide for MINPACK-1.
Applied Math. Division, Argonne National Laboratory, Argonne,
Illinois, Report ANL-80-74, 1980.

```
Numerical Aspects
```  The algorithm requires 0(N*(BSN+ST)) operations and is backward
stable, if R is nonsingular.

```
```  This routine is a structure-exploiting, LAPACK-based modification
of QRSOLV from the MINPACK package , and with optional
condition estimation.
The option COND = 'U' is useful when dealing with several
right-hand side vectors.

```
Example

Program Text

```  None
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Program Data
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Program Results
```  None
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