## NF01BW

### Matrix-vector product x <-- (J' J + c I) x, for J in a compressed form

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

Purpose

```  To compute the matrix-vector product x <-- (J'*J + c*I)*x, for the
Jacobian J as received from SLICOT Library routine NF01BD:

/  dy(1)/dwb(1)  |  dy(1)/dtheta  \
Jc = |       :        |       :        | .
\  dy(L)/dwb(L)  |  dy(L)/dtheta  /

This is a compressed representation of the actual structure

/   J_1    0    ..   0   |  L_1  \
|    0    J_2   ..   0   |  L_2  |
J = |    :     :    ..   :   |   :   | .
|    :     :    ..   :   |   :   |
\    0     0    ..  J_L  |  L_L  /

```
Specification
```      SUBROUTINE NF01BW( N, IPAR, LIPAR, DPAR, LDPAR, J, LDJ, X, INCX,
\$                   DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
INTEGER           INCX, INFO, LDJ, LDPAR, LDWORK, LIPAR, N
C     .. Array Arguments ..
DOUBLE PRECISION  DPAR(*), DWORK(*), J(LDJ,*), X(*)
INTEGER           IPAR(*)

```
Arguments

Input/Output Parameters

```  N       (input) INTEGER
The dimension of the vector x.
N = BN*BSN + ST >= 0.  (See parameter description below.)

IPAR    (input) INTEGER array, dimension (LIPAR)
The integer parameters describing the structure of the
matrix J, as follows:
IPAR(1) must contain ST, the number of parameters
corresponding to the linear part.  ST >= 0.
IPAR(2) must contain BN, the number of blocks, BN = L,
for the parameters corresponding to the nonlinear
part.  BN >= 0.
IPAR(3) must contain BSM, the number of rows of the blocks
J_k = dy(k)/dwb(k), k = 1:BN, if BN > 0, or the
number of rows of the matrix J, if BN <= 1.
IPAR(4) must contain BSN, the number of columns of the
blocks J_k, k = 1:BN.  BSN >= 0.

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

DPAR    (input) DOUBLE PRECISION array, dimension (LDPAR)
The real parameters needed for solving the problem.
The entry DPAR(1) must contain the real scalar c.

LDPAR   (input) INTEGER
The length of the array DPAR.  LDPAR >= 1.

J       (input) DOUBLE PRECISION array, dimension (LDJ, NC)
where NC = N if BN <= 1, and NC = BSN+ST, if BN > 1.
The leading NR-by-NC part of this array must contain
the (compressed) representation (Jc) of the Jacobian
matrix J, where NR = BSM if BN <= 1, and NR = BN*BSM,
if BN > 1.

LDJ     (input) INTEGER
The leading dimension of array J.  LDJ >= MAX(1,NR).

X       (input/output) DOUBLE PRECISION array, dimension
(1+(N-1)*INCX)
On entry, this incremented array must contain the
vector x.
On exit, this incremented array contains the value of the
matrix-vector product (J'*J + c*I)*x.

INCX    (input) INTEGER
The increment for the elements of X.  INCX >= 1.

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (LDWORK)

LDWORK  INTEGER
The length of the array DWORK.  LDWORK >= NR.

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

```
Method
```  The associativity of matrix multiplications is used; the result
is obtained as:  x_out = J'*( J*x ) + c*x.

```
```  None
```
Example

Program Text

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