MD03AD

Solution of a standard nonlinear least squares problem (Cholesky-based or conjugate gradients solver)

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

Purpose

  To minimize the sum of the squares of m nonlinear functions, e, in
  n variables, x, by a modification of the Levenberg-Marquardt
  algorithm, using either a Cholesky-based or a conjugate gradients
  solver. The user must provide a subroutine FCN which calculates
  the functions and the Jacobian J (possibly by finite differences),
  and another subroutine JPJ, which computes either J'*J + par*I
  (if ALG = 'D'), or (J'*J + par*I)*x (if ALG = 'I'), where par is
  the Levenberg factor, exploiting the possible structure of the
  Jacobian matrix. Template implementations of these routines are
  included in the SLICOT Library.

Specification
      SUBROUTINE MD03AD( XINIT, ALG, STOR, UPLO, FCN, JPJ, M, N, ITMAX,
     $                   NPRINT, IPAR, LIPAR, DPAR1, LDPAR1, DPAR2,
     $                   LDPAR2, X, NFEV, NJEV, TOL, CGTOL, DWORK,
     $                   LDWORK, IWARN, INFO )
C     .. Scalar Arguments ..
      CHARACTER         ALG, STOR, UPLO, XINIT
      INTEGER           INFO, ITMAX, IWARN, LDPAR1, LDPAR2, LDWORK,
     $                  LIPAR, M, N, NFEV, NJEV, NPRINT
      DOUBLE PRECISION  CGTOL, TOL
C     .. Array Arguments ..
      DOUBLE PRECISION  DPAR1(LDPAR1,*), DPAR2(LDPAR2,*), DWORK(*), X(*)
      INTEGER           IPAR(*)

Arguments

Mode Parameters

  XINIT   CHARACTER*1
          Specifies how the variables x are initialized, as follows:
          = 'R' :  the array X is initialized to random values; the
                   entries DWORK(1:4) are used to initialize the
                   random number generator: the first three values
                   are converted to integers between 0 and 4095, and
                   the last one is converted to an odd integer
                   between 1 and 4095;
          = 'G' :  the given entries of X are used as initial values
                   of variables.

  ALG     CHARACTER*1
          Specifies the algorithm used for solving the linear
          systems involving a Jacobian matrix J, as follows:
          = 'D' :  a direct algorithm, which computes the Cholesky
                   factor of the matrix J'*J + par*I is used;
          = 'I' :  an iterative Conjugate Gradients algorithm, which
                   only needs the matrix J, is used.
          In both cases, matrix J is stored in a compressed form.

  STOR    CHARACTER*1
          If ALG = 'D', specifies the storage scheme for the
          symmetric matrix J'*J, as follows:
          = 'F' :  full storage is used;
          = 'P' :  packed storage is used.
          The option STOR = 'F' usually ensures a faster execution.
          This parameter is not relevant if ALG = 'I'.

  UPLO    CHARACTER*1
          If ALG = 'D', specifies which part of the matrix J'*J
          is stored, as follows:
          = 'U' :  the upper triagular part is stored;
          = 'L' :  the lower triagular part is stored.
          The option UPLO = 'U' usually ensures a faster execution.
          This parameter is not relevant if ALG = 'I'.

Function Parameters
  FCN     EXTERNAL
          Subroutine which evaluates the functions and the Jacobian.
          FCN must be declared in an external statement in the user
          calling program, and must have the following interface:

          SUBROUTINE FCN( IFLAG, M, N, IPAR, LIPAR, DPAR1, LDPAR1,
         $                DPAR2, LDPAR2, X, NFEVL, E, J, LDJ, JTE,
         $                DWORK, LDWORK, INFO )

          where

          IFLAG   (input/output) INTEGER
                  On entry, this parameter must contain a value
                  defining the computations to be performed:
                  = 0 :  Optionally, print the current iterate X,
                         function values E, and Jacobian matrix J,
                         or other results defined in terms of these
                         values. See the argument NPRINT of MD03AD.
                         Do not alter E and J.
                  = 1 :  Calculate the functions at X and return
                         this vector in E. Do not alter J.
                  = 2 :  Calculate the Jacobian at X and return
                         this matrix in J. Also return J'*e in JTE
                         and NFEVL (see below). Do not alter E.
                  = 3 :  Do not compute neither the functions nor
                         the Jacobian, but return in LDJ and
                         IPAR/DPAR1,DPAR2 (some of) the integer/real
                         parameters needed.
                  On exit, the value of this parameter should not be
                  changed by FCN unless the user wants to terminate
                  execution of MD03AD, in which case IFLAG must be
                  set to a negative integer.

          M       (input) INTEGER
                  The number of functions.  M >= 0.

          N       (input) INTEGER
                  The number of variables.  M >= N >= 0.

          IPAR    (input/output) INTEGER array, dimension (LIPAR)
                  The integer parameters describing the structure of
                  the Jacobian matrix or needed for problem solving.
                  IPAR is an input parameter, except for IFLAG = 3
                  on entry, when it is also an output parameter.
                  On exit, if IFLAG = 3, IPAR(1) contains the length
                  of the array J, for storing the Jacobian matrix,
                  and the entries IPAR(2:5) contain the workspace
                  required by FCN for IFLAG = 1, FCN for IFLAG = 2,
                  JPJ for ALG = 'D', and JPJ for ALG = 'I',
                  respectively.

          LIPAR   (input) INTEGER
                  The length of the array IPAR.  LIPAR >= 5.

          DPAR1   (input/output) DOUBLE PRECISION array, dimension
                  (LDPAR1,*) or (LDPAR1)
                  A first set of real parameters needed for
                  describing or solving the problem.
                  DPAR1 can also be used as an additional array for
                  intermediate results when computing the functions
                  or the Jacobian. For control problems, DPAR1 could
                  store the input trajectory of a system.

          LDPAR1  (input) INTEGER
                  The leading dimension or the length of the array
                  DPAR1, as convenient.  LDPAR1 >= 0.  (LDPAR1 >= 1,
                  if leading dimension.)

          DPAR2   (input/output) DOUBLE PRECISION array, dimension
                  (LDPAR2,*) or (LDPAR2)
                  A second set of real parameters needed for
                  describing or solving the problem.
                  DPAR2 can also be used as an additional array for
                  intermediate results when computing the functions
                  or the Jacobian. For control problems, DPAR2 could
                  store the output trajectory of a system.

          LDPAR2  (input) INTEGER
                  The leading dimension or the length of the array
                  DPAR2, as convenient.  LDPAR2 >= 0.  (LDPAR2 >= 1,
                  if leading dimension.)

          X       (input) DOUBLE PRECISION array, dimension (N)
                  This array must contain the value of the
                  variables x where the functions or the Jacobian
                  must be evaluated.

          NFEVL   (input/output) INTEGER
                  The number of function evaluations needed to
                  compute the Jacobian by a finite difference
                  approximation.
                  NFEVL is an input parameter if IFLAG = 0, or an
                  output parameter if IFLAG = 2. If the Jacobian is
                  computed analytically, NFEVL should be set to a
                  non-positive value.

          E       (input/output) DOUBLE PRECISION array,
                  dimension (M)
                  This array contains the value of the (error)
                  functions e evaluated at X.
                  E is an input parameter if IFLAG = 0 or 2, or an
                  output parameter if IFLAG = 1.

          J       (input/output) DOUBLE PRECISION array, dimension
                  (LDJ,NC), where NC is the number of columns
                  needed.
                  This array contains a possibly compressed
                  representation of the Jacobian matrix evaluated
                  at X. If full Jacobian is stored, then NC = N.
                  J is an input parameter if IFLAG = 0, or an output
                  parameter if IFLAG = 2.

          LDJ     (input/output) INTEGER
                  The leading dimension of array J.  LDJ >= 1.
                  LDJ is essentially used inside the routines FCN
                  and JPJ.
                  LDJ is an input parameter, except for IFLAG = 3
                  on entry, when it is an output parameter.
                  It is assumed in MD03AD that LDJ is not larger
                  than needed.

          JTE     (output) DOUBLE PRECISION array, dimension (N)
                  If IFLAG = 2, the matrix-vector product J'*e.

          DWORK   DOUBLE PRECISION array, dimension (LDWORK)
                  The workspace array for subroutine FCN.
                  On exit, if INFO = 0, DWORK(1) returns the optimal
                  value of LDWORK.

          LDWORK  (input) INTEGER
                  The size of the array DWORK (as large as needed
                  in the subroutine FCN).  LDWORK >= 1.

          INFO    INTEGER
                  Error indicator, set to a negative value if an
                  input (scalar) argument is erroneous, and to
                  positive values for other possible errors in the
                  subroutine FCN. The LAPACK Library routine XERBLA
                  should be used in conjunction with negative INFO.
                  INFO must be zero if the subroutine finished
                  successfully.

          Parameters marked with "(input)" must not be changed.

  JPJ     EXTERNAL
          Subroutine which computes J'*J + par*I, if ALG = 'D', and
          J'*J*x + par*x, if ALG = 'I', where J is the Jacobian as
          described above.

          JPJ must have the following interface:

          SUBROUTINE JPJ( STOR, UPLO, N, IPAR, LIPAR, DPAR, LDPAR,
         $                J, LDJ, JTJ, LDJTJ, DWORK, LDWORK, INFO )

          if ALG = 'D', and

          SUBROUTINE JPJ( N, IPAR, LIPAR, DPAR, LDPAR, J, LDJ, X,
         $                INCX, DWORK, LDWORK, INFO )

          if ALG = 'I', where

          STOR    (input) CHARACTER*1
                  Specifies the storage scheme for the symmetric
                  matrix J'*J, as follows:
                  = 'F' :  full storage is used;
                  = 'P' :  packed storage is used.

          UPLO    (input) CHARACTER*1
                  Specifies which part of the matrix J'*J is stored,
                  as follows:
                  = 'U' :  the upper triagular part is stored;
                  = 'L' :  the lower triagular part is stored.

          N       (input) INTEGER
                  The number of columns of the matrix J.  N >= 0.

          IPAR    (input) INTEGER array, dimension (LIPAR)
                  The integer parameters describing the structure of
                  the Jacobian matrix.

          LIPAR   (input) INTEGER
                  The length of the array IPAR.  LIPAR >= 0.

          DPAR    (input) DOUBLE PRECISION array, dimension (LDPAR)
                  DPAR(1) must contain an initial estimate of the
                  Levenberg-Marquardt parameter, par.  DPAR(1) >= 0.

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

          J       (input) DOUBLE PRECISION array, dimension
                  (LDJ, NC), where NC is the number of columns.
                  The leading NR-by-NC part of this array must
                  contain the (compressed) representation of the
                  Jacobian matrix J, where NR is the number of rows
                  of J (function of IPAR entries).

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

          JTJ     (output) DOUBLE PRECISION array,
                           dimension (LDJTJ,N),    if STOR = 'F',
                           dimension (N*(N+1)/2),  if STOR = 'P'.
                  The leading N-by-N (if STOR = 'F'), or N*(N+1)/2
                  (if STOR = 'P') part of this array contains the
                  upper or lower triangle of the matrix J'*J+par*I,
                  depending on UPLO = 'U', or UPLO = 'L',
                  respectively, stored either as a two-dimensional,
                  or one-dimensional array, depending on STOR.

          LDJTJ   (input) INTEGER
                  The leading dimension of the array JTJ.
                  LDJTJ >= MAX(1,N), if STOR = 'F'.
                  LDJTJ >= 1,        if STOR = 'P'.

          DWORK   DOUBLE PRECISION array, dimension (LDWORK)
                  The workspace array for subroutine JPJ.

          LDWORK  (input) INTEGER
                  The size of the array DWORK (as large as needed
                  in the subroutine JPJ).

          INFO    INTEGER
                  Error indicator, set to a negative value if an
                  input (scalar) argument is erroneous, and to
                  positive values for other possible errors in the
                  subroutine JPJ. The LAPACK Library routine XERBLA
                  should be used in conjunction with negative INFO
                  values. INFO must be zero if the subroutine
                  finished successfully.

          If ALG = 'I', the parameters in common with those for
          ALG = 'D', have the same meaning, and the additional
          parameters are:

          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 + par)*x.

          INCX    (input) INTEGER
                  The increment for the elements of X.  INCX > 0.

          Parameters marked with "(input)" must not be changed.

Input/Output Parameters
  M       (input) INTEGER
          The number of functions.  M >= 0.

  N       (input) INTEGER
          The number of variables.  M >= N >= 0.

  ITMAX   (input) INTEGER
          The maximum number of iterations.  ITMAX >= 0.

  NPRINT  (input) INTEGER
          This parameter enables controlled printing of iterates if
          it is positive. In this case, FCN is called with IFLAG = 0
          at the beginning of the first iteration and every NPRINT
          iterations thereafter and immediately prior to return,
          with X, E, and J available for printing. If NPRINT is not
          positive, no special calls of FCN with IFLAG = 0 are made.

  IPAR    (input) INTEGER array, dimension (LIPAR)
          The integer parameters needed, for instance, for
          describing the structure of the Jacobian matrix, which
          are handed over to the routines FCN and JPJ.
          The first five entries of this array are modified
          internally by a call to FCN (with IFLAG = 3), but are
          restored on exit.

  LIPAR   (input) INTEGER
          The length of the array IPAR.  LIPAR >= 5.

  DPAR1   (input/output) DOUBLE PRECISION array, dimension
          (LDPAR1,*) or (LDPAR1)
          A first set of real parameters needed for describing or
          solving the problem. This argument is not used by MD03AD
          routine, but it is passed to the routine FCN.

  LDPAR1  (input) INTEGER
          The leading dimension or the length of the array DPAR1, as
          convenient.  LDPAR1 >= 0.  (LDPAR1 >= 1, if leading
          dimension.)

  DPAR2   (input/output) DOUBLE PRECISION array, dimension
          (LDPAR2,*) or (LDPAR2)
          A second set of real parameters needed for describing or
          solving the problem. This argument is not used by MD03AD
          routine, but it is passed to the routine FCN.

  LDPAR2  (input) INTEGER
          The leading dimension or the length of the array DPAR2, as
          convenient.  LDPAR2 >= 0.  (LDPAR2 >= 1, if leading
          dimension.)

  X       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, if XINIT = 'G', this array must contain the
          vector of initial variables x to be optimized.
          If XINIT = 'R', this array need not be set before entry,
          and random values will be used to initialize x.
          On exit, if INFO = 0, this array contains the vector of
          values that (approximately) minimize the sum of squares of
          error functions. The values returned in IWARN and
          DWORK(1:5) give details on the iterative process.

  NFEV    (output) INTEGER
          The number of calls to FCN with IFLAG = 1. If FCN is
          properly implemented, this includes the function
          evaluations needed for finite difference approximation
          of the Jacobian.

  NJEV    (output) INTEGER
          The number of calls to FCN with IFLAG = 2.

Tolerances
  TOL     DOUBLE PRECISION
          If TOL >= 0, the tolerance which measures the relative
          error desired in the sum of squares. Termination occurs
          when the actual relative reduction in the sum of squares
          is at most TOL. If the user sets  TOL < 0, then  SQRT(EPS)
          is used instead TOL, where EPS is the machine precision
          (see LAPACK Library routine DLAMCH).

  CGTOL   DOUBLE PRECISION
          If ALG = 'I' and CGTOL > 0, the tolerance which measures
          the relative residual of the solutions computed by the
          conjugate gradients (CG) algorithm. Termination of a
          CG process occurs when the relative residual is at
          most CGTOL. If the user sets  CGTOL <= 0, then  SQRT(EPS)
          is used instead CGTOL.

Workspace
  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if INFO = 0, DWORK(1) returns the optimal value
          of LDWORK, DWORK(2) returns the residual error norm (the
          sum of squares), DWORK(3) returns the number of iterations
          performed, DWORK(4) returns the total number of conjugate
          gradients iterations performed (zero, if ALG = 'D'), and
          DWORK(5) returns the final Levenberg factor.

  LDWORK  INTEGER
          The length of the array DWORK.
          LDWORK >= max( 5, M + 2*N + size(J) +
                         max( DW( FCN|IFLAG = 1 ) + N,
                              DW( FCN|IFLAG = 2 ),
                              DW( sol ) ) ),
          where size(J) is the size of the Jacobian (provided by FCN
          in IPAR(1), for IFLAG = 3), DW( f ) is the workspace
          needed by the routine f, where f is FCN or JPJ (provided
          by FCN in IPAR(2:5), for IFLAG = 3), and DW( sol ) is the
          workspace needed for solving linear systems,
          DW( sol ) = N*N + DW( JPJ ),  if ALG = 'D', STOR = 'F';
          DW( sol ) = N*(N+1)/2 + DW( JPJ ),
                                        if ALG = 'D', STOR = 'P';
          DW( sol ) = 3*N + DW( JPJ ),  if ALG = 'I'.

Warning Indicator
  IWARN   INTEGER
          < 0:  the user set IFLAG = IWARN in the subroutine FCN;
          = 0:  no warning;
          = 1:  if the iterative process did not converge in ITMAX
                iterations with tolerance TOL;
          = 2:  if ALG = 'I', and in one or more iterations of the
                Levenberg-Marquardt algorithm, the conjugate
                gradient algorithm did not finish after 3*N
                iterations, with the accuracy required in the
                call;
          = 3:  the cosine of the angle between e and any column of
                the Jacobian is at most FACTOR*EPS in absolute
                value, where FACTOR = 100 is defined in a PARAMETER
                statement;
          = 4:  TOL is too small: no further reduction in the sum
                of squares is possible.
                In all these cases, DWORK(1:5) are set as described
                above.

Error Indicator
  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          = 1:  user-defined routine FCN returned with INFO <> 0
                for IFLAG = 1;
          = 2:  user-defined routine FCN returned with INFO <> 0
                for IFLAG = 2;
          = 3:  SLICOT Library routine MB02XD, if ALG = 'D', or
                SLICOT Library routine MB02WD, if ALG = 'I' (or
                user-defined routine JPJ), returned with INFO <> 0.

Method
  If XINIT = 'R', the initial value for X is set to a vector of
  pseudo-random values uniformly distributed in [-1,1].

  The Levenberg-Marquardt algorithm (described in [1]) is used for
  optimizing the parameters. This algorithm needs the Jacobian
  matrix J, which is provided by the subroutine FCN. The algorithm
  tries to update x by the formula

      x = x - p,

  using the solution of the system of linear equations

      (J'*J + PAR*I)*p = J'*e,

  where I is the identity matrix, and e the error function vector.
  The Levenberg factor PAR is decreased after each successfull step
  and increased in the other case.

  If ALG = 'D', a direct method, which evaluates the matrix product
  J'*J + par*I and then factors it using Cholesky algorithm,
  implemented in the SLICOT Libray routine MB02XD, is used for
  solving the linear system above.

  If ALG = 'I', the Conjugate Gradients method, described in [2],
  and implemented in the SLICOT Libray routine MB02WD, is used for
  solving the linear system above. The main advantage of this method
  is that in most cases the solution of the system can be computed
  in less time than the time needed to compute the matrix J'*J
  This is, however, problem dependent.

References
  [1] Kelley, C.T.
      Iterative Methods for Optimization.
      Society for Industrial and Applied Mathematics (SIAM),
      Philadelphia (Pa.), 1999.

  [2] Golub, G.H. and van Loan, C.F.
      Matrix Computations. Third Edition.
      M. D. Johns Hopkins University Press, Baltimore, pp. 520-528,
      1996.

  [3] More, J.J.
      The Levenberg-Marquardt algorithm: implementation and theory.
      In Watson, G.A. (Ed.), Numerical Analysis, Lecture Notes in
      Mathematics, vol. 630, Springer-Verlag, Berlin, Heidelberg
      and New York, pp. 105-116, 1978.

Numerical Aspects
  The Levenberg-Marquardt algorithm described in [3] is scaling
  invariant and globally convergent to (maybe local) minima.
  According to [1], the convergence rate near a local minimum is
  quadratic, if the Jacobian is computed analytically, and linear,
  if the Jacobian is computed numerically.

  Whether or not the direct algorithm is faster than the iterative
  Conjugate Gradients algorithm for solving the linear systems
  involved depends on several factors, including the conditioning
  of the Jacobian matrix, and the ratio between its dimensions.

Further Comments
  None
Example

Program Text

*     MD03AD 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           LDWORK
      PARAMETER         ( LDWORK = MMAX + 2*NMAX + MMAX*NMAX +
     $                             MAX( NMAX*NMAX, 3*NMAX + MMAX ) )
*     .. The lengths of DPAR1, DPAR2, IPAR are set to 1, 1, and 5 ..
      INTEGER           LDPAR1, LDPAR2, LIPAR
      PARAMETER         ( LDPAR1 = 1, LDPAR2 = 1, LIPAR = 5 )
*     .. Local Scalars ..
      CHARACTER*1       ALG, STOR, UPLO, XINIT
      INTEGER           I, INFO, ITMAX, IWARN, M, N, NFEV, NJEV, NPRINT
      DOUBLE PRECISION  CGTOL, TOL
*     .. Array Arguments ..
      INTEGER           IPAR(LIPAR)
      DOUBLE PRECISION  DPAR1(LDPAR1), DPAR2(LDPAR2), DWORK(LDWORK),
     $                  X(NMAX)
*     .. External Functions ..
      LOGICAL           LSAME
      EXTERNAL          LSAME
*     .. External Subroutines ..
      EXTERNAL          MD03AD, MD03AF, NF01BV, NF01BX
*     .. 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, ITMAX, NPRINT, TOL, CGTOL, XINIT,
     $                      ALG, STOR, UPLO
      IF( M.LE.0 .OR. M.GT.MMAX ) THEN
         WRITE ( NOUT, FMT = 99993 ) M
      ELSE
         IF( N.LE.0 .OR. N.GT.NMAX ) THEN
            WRITE ( NOUT, FMT = 99992 ) N
         ELSE
            IF ( LSAME( XINIT, 'G' ) )
     $         READ ( NIN, FMT = * ) ( X(I), I = 1,N )
*           Solve a standard nonlinear least squares problem.
            IPAR(1) = M
            IF ( LSAME( ALG, 'D' ) ) THEN
               CALL MD03AD( XINIT, ALG, STOR, UPLO, MD03AF, NF01BV, M,
     $                      N, ITMAX, NPRINT, IPAR, LIPAR, DPAR1,
     $                      LDPAR1, DPAR2, LDPAR2, X, NFEV, NJEV, TOL,
     $                      CGTOL, DWORK, LDWORK, IWARN, INFO )
            ELSE
               CALL MD03AD( XINIT, ALG, STOR, UPLO, MD03AF, NF01BX, M,
     $                      N, ITMAX, NPRINT, IPAR, LIPAR, DPAR1,
     $                      LDPAR1, DPAR2, LDPAR2, X, NFEV, NJEV, TOL,
     $                      CGTOL, DWORK, LDWORK, IWARN, INFO )
            END IF
*
            IF ( INFO.NE.0 ) THEN
               WRITE ( NOUT, FMT = 99998 ) INFO
            ELSE
               IF( IWARN.NE.0 ) WRITE ( NOUT, FMT = 99991 ) IWARN
               WRITE ( NOUT, FMT = 99997 ) DWORK(2)
               WRITE ( NOUT, FMT = 99996 ) NFEV, NJEV
               WRITE ( NOUT, FMT = 99994 )
               WRITE ( NOUT, FMT = 99995 ) ( X(I), I = 1, N )
            END IF
         END IF
      END IF
      STOP
*
99999 FORMAT (' MD03AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MD03AD = ',I2)
99997 FORMAT (/' Final 2-norm of the residuals = ',D15.7)
99996 FORMAT (/' The number of function and Jacobian evaluations = ',
     $           2I7)
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' Final approximate solution is ' )
99993 FORMAT (/' M is out of range.',/' M = ',I5)
99992 FORMAT (/' N is out of range.',/' N = ',I5)
99991 FORMAT (' IWARN on exit from MD03AD = ',I2)
      END
C
      SUBROUTINE MD03AF( IFLAG, M, N, IPAR, LIPAR, DPAR1, LDPAR1, DPAR2,
     $                   LDPAR2, X, NFEVL, E, J, LDJ, JTE, DWORK,
     $                   LDWORK, INFO )
C
C     This is the FCN routine for solving a standard nonlinear least
C     squares problem using SLICOT Library routine MD03AD. See the
C     argument FCN in the routine MD03AD for the description of
C     parameters.
C
C     The example programmed in this routine is adapted from that
C     accompanying the MINPACK routine LMDER.
C
C     ******************************************************************
C
C     .. Parameters ..
C     .. NOUT is the unit number for printing intermediate results ..
      INTEGER           NOUT
      PARAMETER         ( NOUT = 6 )
      DOUBLE PRECISION  ZERO, ONE
      PARAMETER         ( ZERO = 0.0D0, ONE = 1.0D0 )
C     .. Scalar Arguments ..
      INTEGER           IFLAG, INFO, LDJ, LDPAR1, LDPAR2, LDWORK, LIPAR,
     $                  M, N, NFEVL
C     .. Array Arguments ..
      INTEGER           IPAR(*)
      DOUBLE PRECISION  DPAR1(*), DPAR2(*), DWORK(*), E(*), J(LDJ,*),
     $                  JTE(*), X(*)
C     .. Local Scalars ..
      INTEGER           I
      DOUBLE PRECISION  ERR, TMP1, TMP2, TMP3, TMP4
C     .. External Functions ..
      DOUBLE PRECISION  DNRM2
      EXTERNAL          DNRM2
C     .. External Subroutines ..
      EXTERNAL          DGEMV
C     .. DATA Statements ..
      DOUBLE PRECISION  Y(15)
      DATA              Y(1), Y(2), Y(3), Y(4), Y(5), Y(6), Y(7), Y(8),
     $                  Y(9), Y(10), Y(11), Y(12), Y(13), Y(14), Y(15)
     $                  / 1.4D-1, 1.8D-1, 2.2D-1, 2.5D-1, 2.9D-1,
     $                    3.2D-1, 3.5D-1, 3.9D-1, 3.7D-1, 5.8D-1,
     $                    7.3D-1, 9.6D-1, 1.34D0, 2.1D0,  4.39D0 /
C
C     .. Executable Statements ..
C
      INFO = 0
      IF ( IFLAG.EQ.1 ) THEN
C
C        Compute the error function values, e.
C
         DO 10 I = 1, 15
            TMP1 = I
            TMP2 = 16 - I
            IF ( I.GT.8 ) THEN
               TMP3 = TMP2
            ELSE
               TMP3 = TMP1
            END IF
            E(I) = Y(I) - ( X(1) + TMP1/( X(2)*TMP2 + X(3)*TMP3 ) )
   10    CONTINUE
C
      ELSE IF ( IFLAG.EQ.2 ) THEN
C
C        Compute the Jacobian.
C
         DO 30 I = 1, 15
            TMP1 = I
            TMP2 = 16 - I
            IF ( I.GT.8 ) THEN
               TMP3 = TMP2
            ELSE
               TMP3 = TMP1
            END IF
            TMP4 = ( X(2)*TMP2 + X(3)*TMP3 )**2
            J(I,1) = -ONE
            J(I,2) = TMP1*TMP2/TMP4
            J(I,3) = TMP1*TMP3/TMP4
   30    CONTINUE
C
C        Compute the product J'*e (the error e was computed in array E).
C
         CALL DGEMV( 'Transpose', M, N, ONE, J, LDJ, E, 1, ZERO, JTE,
     $               1 )
C
         NFEVL = 0
C
      ELSE IF ( IFLAG.EQ.3 ) THEN
C
C        Set the parameter LDJ, the length of the array J, and the sizes
C        of the workspace for MD03AF (IFLAG = 1 or 2), NF01BV and
C        NF01BX.
C
         LDJ = M
         IPAR(1) = M*N
         IPAR(2) = 0
         IPAR(3) = 0
         IPAR(4) = M
      ELSE IF ( IFLAG.EQ.0 ) THEN
C
C        Special call for printing intermediate results.
C
         ERR = DNRM2( M, E, 1 )
         WRITE( NOUT, '('' Norm of current error = '', D15.6)') ERR
C
      END IF
C
      DWORK(1) = ZERO
      RETURN
C
C *** Last line of MD03AF ***
      END
Program Data
 MD03AD EXAMPLE PROGRAM DATA
 15     3   100     0   -1.   -1.    G     D     F    U
   1.0   1.0   1.0
Program Results
 MD03AD EXAMPLE PROGRAM RESULTS


 Final 2-norm of the residuals =   0.9063596D-01

 The number of function and Jacobian evaluations =      13     12

 Final approximate solution is 
   0.0824   1.1330   2.3437

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