*DECK DWNNLS SUBROUTINE DWNNLS (W, MDW, ME, MA, N, L, PRGOPT, X, RNORM, MODE, + IWORK, WORK) C***BEGIN PROLOGUE DWNNLS C***PURPOSE Solve a linearly constrained least squares problem with C equality constraints and nonnegativity constraints on C selected variables. C***LIBRARY SLATEC C***CATEGORY K1A2A C***TYPE DOUBLE PRECISION (WNNLS-S, DWNNLS-D) C***KEYWORDS CONSTRAINED LEAST SQUARES, CURVE FITTING, DATA FITTING, C EQUALITY CONSTRAINTS, INEQUALITY CONSTRAINTS, C NONNEGATIVITY CONSTRAINTS, QUADRATIC PROGRAMMING C***AUTHOR Hanson, R. J., (SNLA) C Haskell, K. H., (SNLA) C***DESCRIPTION C C Abstract C C This subprogram solves a linearly constrained least squares C problem. Suppose there are given matrices E and A of C respective dimensions ME by N and MA by N, and vectors F C and B of respective lengths ME and MA. This subroutine C solves the problem C C EX = F, (equations to be exactly satisfied) C C AX = B, (equations to be approximately satisfied, C in the least squares sense) C C subject to components L+1,...,N nonnegative C C Any values ME.GE.0, MA.GE.0 and 0.LE. L .LE.N are permitted. C C The problem is reposed as problem DWNNLS C C (WT*E)X = (WT*F) C ( A) ( B), (least squares) C subject to components L+1,...,N nonnegative. C C The subprogram chooses the heavy weight (or penalty parameter) WT. C C The parameters for DWNNLS are C C INPUT.. All TYPE REAL variables are DOUBLE PRECISION C C W(*,*),MDW, The array W(*,*) is double subscripted with first C ME,MA,N,L dimensioning parameter equal to MDW. For this C discussion let us call M = ME + MA. Then MDW C must satisfy MDW.GE.M. The condition MDW.LT.M C is an error. C C The array W(*,*) contains the matrices and vectors C C (E F) C (A B) C C in rows and columns 1,...,M and 1,...,N+1 C respectively. Columns 1,...,L correspond to C unconstrained variables X(1),...,X(L). The C remaining variables are constrained to be C nonnegative. The condition L.LT.0 or L.GT.N is C an error. C C PRGOPT(*) This double precision array is the option vector. C If the user is satisfied with the nominal C subprogram features set C C PRGOPT(1)=1 (or PRGOPT(1)=1.0) C C Otherwise PRGOPT(*) is a linked list consisting of C groups of data of the following form C C LINK C KEY C DATA SET C C The parameters LINK and KEY are each one word. C The DATA SET can be comprised of several words. C The number of items depends on the value of KEY. C The value of LINK points to the first C entry of the next group of data within C PRGOPT(*). The exception is when there are C no more options to change. In that C case LINK=1 and the values KEY and DATA SET C are not referenced. The general layout of C PRGOPT(*) is as follows. C C ...PRGOPT(1)=LINK1 (link to first entry of next group) C . PRGOPT(2)=KEY1 (key to the option change) C . PRGOPT(3)=DATA VALUE (data value for this change) C . . C . . C . . C ...PRGOPT(LINK1)=LINK2 (link to the first entry of C . next group) C . PRGOPT(LINK1+1)=KEY2 (key to the option change) C . PRGOPT(LINK1+2)=DATA VALUE C ... . C . . C . . C ...PRGOPT(LINK)=1 (no more options to change) C C Values of LINK that are nonpositive are errors. C A value of LINK.GT.NLINK=100000 is also an error. C This helps prevent using invalid but positive C values of LINK that will probably extend C beyond the program limits of PRGOPT(*). C Unrecognized values of KEY are ignored. The C order of the options is arbitrary and any number C of options can be changed with the following C restriction. To prevent cycling in the C processing of the option array a count of the C number of options changed is maintained. C Whenever this count exceeds NOPT=1000 an error C message is printed and the subprogram returns. C C OPTIONS.. C C KEY=6 C Scale the nonzero columns of the C entire data matrix C (E) C (A) C to have length one. The DATA SET for C this option is a single value. It must C be nonzero if unit length column scaling is C desired. C C KEY=7 C Scale columns of the entire data matrix C (E) C (A) C with a user-provided diagonal matrix. C The DATA SET for this option consists C of the N diagonal scaling factors, one for C each matrix column. C C KEY=8 C Change the rank determination tolerance from C the nominal value of SQRT(SRELPR). This quantity C can be no smaller than SRELPR, The arithmetic- C storage precision. The quantity used C here is internally restricted to be at C least SRELPR. The DATA SET for this option C is the new tolerance. C C KEY=9 C Change the blow-up parameter from the C nominal value of SQRT(SRELPR). The reciprocal of C this parameter is used in rejecting solution C components as too large when a variable is C first brought into the active set. Too large C means that the proposed component times the C reciprocal of the parameter is not less than C the ratio of the norms of the right-side C vector and the data matrix. C This parameter can be no smaller than SRELPR, C the arithmetic-storage precision. C C For example, suppose we want to provide C a diagonal matrix to scale the problem C matrix and change the tolerance used for C determining linear dependence of dropped col C vectors. For these options the dimensions of C PRGOPT(*) must be at least N+6. The FORTRAN C statements defining these options would C be as follows. C C PRGOPT(1)=N+3 (link to entry N+3 in PRGOPT(*)) C PRGOPT(2)=7 (user-provided scaling key) C C CALL DCOPY(N,D,1,PRGOPT(3),1) (copy the N C scaling factors from a user array called D(*) C into PRGOPT(3)-PRGOPT(N+2)) C C PRGOPT(N+3)=N+6 (link to entry N+6 of PRGOPT(*)) C PRGOPT(N+4)=8 (linear dependence tolerance key) C PRGOPT(N+5)=... (new value of the tolerance) C C PRGOPT(N+6)=1 (no more options to change) C C C IWORK(1), The amounts of working storage actually allocated C IWORK(2) for the working arrays WORK(*) and IWORK(*), C respectively. These quantities are compared with C the actual amounts of storage needed for DWNNLS( ). C Insufficient storage allocated for either WORK(*) C or IWORK(*) is considered an error. This feature C was included in DWNNLS( ) because miscalculating C the storage formulas for WORK(*) and IWORK(*) C might very well lead to subtle and hard-to-find C execution errors. C C The length of WORK(*) must be at least C C LW = ME+MA+5*N C This test will not be made if IWORK(1).LE.0. C C The length of IWORK(*) must be at least C C LIW = ME+MA+N C This test will not be made if IWORK(2).LE.0. C C OUTPUT.. All TYPE REAL variables are DOUBLE PRECISION C C X(*) An array dimensioned at least N, which will C contain the N components of the solution vector C on output. C C RNORM The residual norm of the solution. The value of C RNORM contains the residual vector length of the C equality constraints and least squares equations. C C MODE The value of MODE indicates the success or failure C of the subprogram. C C MODE = 0 Subprogram completed successfully. C C = 1 Max. number of iterations (equal to C 3*(N-L)) exceeded. Nearly all problems C should complete in fewer than this C number of iterations. An approximate C solution and its corresponding residual C vector length are in X(*) and RNORM. C C = 2 Usage error occurred. The offending C condition is noted with the error C processing subprogram, XERMSG( ). C C User-designated C Working arrays.. C C WORK(*) A double precision working array of length at least C M + 5*N. C C IWORK(*) An integer-valued working array of length at least C M+N. C C***REFERENCES K. H. Haskell and R. J. Hanson, An algorithm for C linear least squares problems with equality and C nonnegativity constraints, Report SAND77-0552, Sandia C Laboratories, June 1978. C K. H. Haskell and R. J. Hanson, Selected algorithms for C the linearly constrained least squares problem - a C users guide, Report SAND78-1290, Sandia Laboratories, C August 1979. C K. H. Haskell and R. J. Hanson, An algorithm for C linear least squares problems with equality and C nonnegativity constraints, Mathematical Programming C 21 (1981), pp. 98-118. C R. J. Hanson and K. H. Haskell, Two algorithms for the C linearly constrained least squares problem, ACM C Transactions on Mathematical Software, September 1982. C C. L. Lawson and R. J. Hanson, Solving Least Squares C Problems, Prentice-Hall, Inc., 1974. C***ROUTINES CALLED DWNLSM, XERMSG C***REVISION HISTORY (YYMMDD) C 790701 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890618 Completely restructured and revised. (WRB & RWC) C 891006 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ) C 900510 Convert XERRWV calls to XERMSG calls, change Prologue C comments to agree with WNNLS. (RWC) C 920501 Reformatted the REFERENCES section. (WRB) C***END PROLOGUE DWNNLS