*DECK DBOLSM
SUBROUTINE DBOLSM (W, MDW, MINPUT, NCOLS, BL, BU, IND, IOPT, X,
+ RNORM, MODE, RW, WW, SCL, IBASIS, IBB)
C***BEGIN PROLOGUE DBOLSM
C***SUBSIDIARY
C***PURPOSE Subsidiary to DBOCLS and DBOLS
C***LIBRARY SLATEC
C***TYPE DOUBLE PRECISION (SBOLSM-S, DBOLSM-D)
C***AUTHOR (UNKNOWN)
C***DESCRIPTION
C
C **** Double Precision Version of SBOLSM ****
C **** All INPUT and OUTPUT real variables are DOUBLE PRECISION ****
C
C Solve E*X = F (least squares sense) with bounds on
C selected X values.
C The user must have DIMENSION statements of the form:
C
C DIMENSION W(MDW,NCOLS+1), BL(NCOLS), BU(NCOLS),
C * X(NCOLS+NX), RW(NCOLS), WW(NCOLS), SCL(NCOLS)
C INTEGER IND(NCOLS), IOPT(1+NI), IBASIS(NCOLS), IBB(NCOLS)
C
C (Here NX=number of extra locations required for options 1,...,7;
C NX=0 for no options; here NI=number of extra locations possibly
C required for options 1-7; NI=0 for no options; NI=14 if all the
C options are simultaneously in use.)
C
C INPUT
C -----
C
C --------------------
C W(MDW,*),MINPUT,NCOLS
C --------------------
C The array W(*,*) contains the matrix [E:F] on entry. The matrix
C [E:F] has MINPUT rows and NCOLS+1 columns. This data is placed in
C the array W(*,*) with E occupying the first NCOLS columns and the
C right side vector F in column NCOLS+1. The row dimension, MDW, of
C the array W(*,*) must satisfy the inequality MDW .ge. MINPUT.
C Other values of MDW are errors. The values of MINPUT and NCOLS
C must be positive. Other values are errors.
C
C ------------------
C BL(*),BU(*),IND(*)
C ------------------
C These arrays contain the information about the bounds that the
C solution values are to satisfy. The value of IND(J) tells the
C type of bound and BL(J) and BU(J) give the explicit values for
C the respective upper and lower bounds.
C
C 1. For IND(J)=1, require X(J) .ge. BL(J).
C 2. For IND(J)=2, require X(J) .le. BU(J).
C 3. For IND(J)=3, require X(J) .ge. BL(J) and
C X(J) .le. BU(J).
C 4. For IND(J)=4, no bounds on X(J) are required.
C The values of BL(*),BL(*) are modified by the subprogram. Values
C other than 1,2,3 or 4 for IND(J) are errors. In the case IND(J)=3
C (upper and lower bounds) the condition BL(J) .gt. BU(J) is an
C error.
C
C -------
C IOPT(*)
C -------
C This is the array where the user can specify nonstandard options
C for DBOLSM. Most of the time this feature can be ignored by
C setting the input value IOPT(1)=99. Occasionally users may have
C needs that require use of the following subprogram options. For
C details about how to use the options see below: IOPT(*) CONTENTS.
C
C Option Number Brief Statement of Purpose
C ----- ------ ----- --------- -- -------
C 1 Move the IOPT(*) processing pointer.
C 2 Change rank determination tolerance.
C 3 Change blow-up factor that determines the
C size of variables being dropped from active
C status.
C 4 Reset the maximum number of iterations to use
C in solving the problem.
C 5 The data matrix is triangularized before the
C problem is solved whenever (NCOLS/MINPUT) .lt.
C FAC. Change the value of FAC.
C 6 Redefine the weighting matrix used for
C linear independence checking.
C 7 Debug output is desired.
C 99 No more options to change.
C
C ----
C X(*)
C ----
C This array is used to pass data associated with options 1,2,3 and
C 5. Ignore this input parameter if none of these options are used.
C Otherwise see below: IOPT(*) CONTENTS.
C
C ----------------
C IBASIS(*),IBB(*)
C ----------------
C These arrays must be initialized by the user. The values
C IBASIS(J)=J, J=1,...,NCOLS
C IBB(J) =1, J=1,...,NCOLS
C are appropriate except when using nonstandard features.
C
C ------
C SCL(*)
C ------
C This is the array of scaling factors to use on the columns of the
C matrix E. These values must be defined by the user. To suppress
C any column scaling set SCL(J)=1.0, J=1,...,NCOLS.
C
C OUTPUT
C ------
C
C ----------
C X(*),RNORM
C ----------
C The array X(*) contains a solution (if MODE .ge. 0 or .eq. -22)
C for the constrained least squares problem. The value RNORM is the
C minimum residual vector length.
C
C ----
C MODE
C ----
C The sign of mode determines whether the subprogram has completed
C normally, or encountered an error condition or abnormal status.
C A value of MODE .ge. 0 signifies that the subprogram has completed
C normally. The value of MODE (.ge. 0) is the number of variables
C in an active status: not at a bound nor at the value ZERO, for
C the case of free variables. A negative value of MODE will be one
C of the 18 cases -38,-37,...,-22, or -1. Values .lt. -1 correspond
C to an abnormal completion of the subprogram. To understand the
C abnormal completion codes see below: ERROR MESSAGES for DBOLSM
C An approximate solution will be returned to the user only when
C maximum iterations is reached, MODE=-22.
C
C -----------
C RW(*),WW(*)
C -----------
C These are working arrays each with NCOLS entries. The array RW(*)
C contains the working (scaled, nonactive) solution values. The
C array WW(*) contains the working (scaled, active) gradient vector
C values.
C
C ----------------
C IBASIS(*),IBB(*)
C ----------------
C These arrays contain information about the status of the solution
C when MODE .ge. 0. The indices IBASIS(K), K=1,...,MODE, show the
C nonactive variables; indices IBASIS(K), K=MODE+1,..., NCOLS are
C the active variables. The value (IBB(J)-1) is the number of times
C variable J was reflected from its upper bound. (Normally the user
C can ignore these parameters.)
C
C IOPT(*) CONTENTS
C ------- --------
C The option array allows a user to modify internal variables in
C the subprogram without recompiling the source code. A central
C goal of the initial software design was to do a good job for most
C people. Thus the use of options will be restricted to a select
C group of users. The processing of the option array proceeds as
C follows: a pointer, here called LP, is initially set to the value
C 1. The value is updated as the options are processed. At the
C pointer position the option number is extracted and used for
C locating other information that allows for options to be changed.
C The portion of the array IOPT(*) that is used for each option is
C fixed; the user and the subprogram both know how many locations
C are needed for each option. A great deal of error checking is
C done by the subprogram on the contents of the option array.
C Nevertheless it is still possible to give the subprogram optional
C input that is meaningless. For example, some of the options use
C the location X(NCOLS+IOFF) for passing data. The user must manage
C the allocation of these locations when more than one piece of
C option data is being passed to the subprogram.
C
C 1
C -
C Move the processing pointer (either forward or backward) to the
C location IOPT(LP+1). The processing pointer is moved to location
C LP+2 of IOPT(*) in case IOPT(LP)=-1. For example to skip over
C locations 3,...,NCOLS+2 of IOPT(*),
C
C IOPT(1)=1
C IOPT(2)=NCOLS+3
C (IOPT(I), I=3,...,NCOLS+2 are not defined here.)
C IOPT(NCOLS+3)=99
C CALL DBOLSM
C
C CAUTION: Misuse of this option can yield some very hard-to-find
C bugs. Use it with care.
C
C 2
C -
C The algorithm that solves the bounded least squares problem
C iteratively drops columns from the active set. This has the
C effect of joining a new column vector to the QR factorization of
C the rectangular matrix consisting of the partially triangularized
C nonactive columns. After triangularizing this matrix a test is
C made on the size of the pivot element. The column vector is
C rejected as dependent if the magnitude of the pivot element is
C .le. TOL* magnitude of the column in components strictly above
C the pivot element. Nominally the value of this (rank) tolerance
C is TOL = SQRT(R1MACH(4)). To change only the value of TOL, for
C example,
C
C X(NCOLS+1)=TOL
C IOPT(1)=2
C IOPT(2)=1
C IOPT(3)=99
C CALL DBOLSM
C
C Generally, if LP is the processing pointer for IOPT(*),
C
C X(NCOLS+IOFF)=TOL
C IOPT(LP)=2
C IOPT(LP+1)=IOFF
C .
C CALL DBOLSM
C
C The required length of IOPT(*) is increased by 2 if option 2 is
C used; The required length of X(*) is increased by 1. A value of
C IOFF .le. 0 is an error. A value of TOL .le. R1MACH(4) gives a
C warning message; it is not considered an error.
C
C 3
C -
C A solution component is left active (not used) if, roughly
C speaking, it seems too large. Mathematically the new component is
C left active if the magnitude is .ge.((vector norm of F)/(matrix
C norm of E))/BLOWUP. Nominally the factor BLOWUP = SQRT(R1MACH(4)).
C To change only the value of BLOWUP, for example,
C
C X(NCOLS+2)=BLOWUP
C IOPT(1)=3
C IOPT(2)=2
C IOPT(3)=99
C CALL DBOLSM
C
C Generally, if LP is the processing pointer for IOPT(*),
C
C X(NCOLS+IOFF)=BLOWUP
C IOPT(LP)=3
C IOPT(LP+1)=IOFF
C .
C CALL DBOLSM
C
C The required length of IOPT(*) is increased by 2 if option 3 is
C used; the required length of X(*) is increased by 1. A value of
C IOFF .le. 0 is an error. A value of BLOWUP .le. 0.0 is an error.
C
C 4
C -
C Normally the algorithm for solving the bounded least squares
C problem requires between NCOLS/3 and NCOLS drop-add steps to
C converge. (this remark is based on examining a small number of
C test cases.) The amount of arithmetic for such problems is
C typically about twice that required for linear least squares if
C there are no bounds and if plane rotations are used in the
C solution method. Convergence of the algorithm, while
C mathematically certain, can be much slower than indicated. To
C avoid this potential but unlikely event ITMAX drop-add steps are
C permitted. Nominally ITMAX=5*(MAX(MINPUT,NCOLS)). To change the
C value of ITMAX, for example,
C
C IOPT(1)=4
C IOPT(2)=ITMAX
C IOPT(3)=99
C CALL DBOLSM
C
C Generally, if LP is the processing pointer for IOPT(*),
C
C IOPT(LP)=4
C IOPT(LP+1)=ITMAX
C .
C CALL DBOLSM
C
C The value of ITMAX must be .gt. 0. Other values are errors. Use
C of this option increases the required length of IOPT(*) by 2.
C
C 5
C -
C For purposes of increased efficiency the MINPUT by NCOLS+1 data
C matrix [E:F] is triangularized as a first step whenever MINPUT
C satisfies FAC*MINPUT .gt. NCOLS. Nominally FAC=0.75. To change the
C value of FAC,
C
C X(NCOLS+3)=FAC
C IOPT(1)=5
C IOPT(2)=3
C IOPT(3)=99
C CALL DBOLSM
C
C Generally, if LP is the processing pointer for IOPT(*),
C
C X(NCOLS+IOFF)=FAC
C IOPT(LP)=5
C IOPT(LP+1)=IOFF
C .
C CALL DBOLSM
C
C The value of FAC must be nonnegative. Other values are errors.
C Resetting FAC=0.0 suppresses the initial triangularization step.
C Use of this option increases the required length of IOPT(*) by 2;
C The required length of of X(*) is increased by 1.
C
C 6
C -
C The norm used in testing the magnitudes of the pivot element
C compared to the mass of the column above the pivot line can be
C changed. The type of change that this option allows is to weight
C the components with an index larger than MVAL by the parameter
C WT. Normally MVAL=0 and WT=1. To change both the values MVAL and
C WT, where LP is the processing pointer for IOPT(*),
C
C X(NCOLS+IOFF)=WT
C IOPT(LP)=6
C IOPT(LP+1)=IOFF
C IOPT(LP+2)=MVAL
C
C Use of this option increases the required length of IOPT(*) by 3.
C The length of X(*) is increased by 1. Values of MVAL must be
C nonnegative and not greater than MINPUT. Other values are errors.
C The value of WT must be positive. Any other value is an error. If
C either error condition is present a message will be printed.
C
C 7
C -
C Debug output, showing the detailed add-drop steps for the
C constrained least squares problem, is desired. This option is
C intended to be used to locate suspected bugs.
C
C 99
C --
C There are no more options to change.
C
C The values for options are 1,...,7,99, and are the only ones
C permitted. Other values are errors. Options -99,-1,...,-7 mean
C that the repective options 99,1,...,7 are left at their default
C values. An example is the option to modify the (rank) tolerance:
C
C X(NCOLS+1)=TOL
C IOPT(1)=-2
C IOPT(2)=1
C IOPT(3)=99
C
C Error Messages for DBOLSM
C ----- -------- --- ---------
C -22 MORE THAN ITMAX = ... ITERATIONS SOLVING BOUNDED LEAST
C SQUARES PROBLEM.
C
C -23 THE OPTION NUMBER = ... IS NOT DEFINED.
C
C -24 THE OFFSET = ... BEYOND POSTION NCOLS = ... MUST BE POSITIVE
C FOR OPTION NUMBER 2.
C
C -25 THE TOLERANCE FOR RANK DETERMINATION = ... IS LESS THAN
C MACHINE PRECISION = ....
C
C -26 THE OFFSET = ... BEYOND POSITION NCOLS = ... MUST BE POSTIVE
C FOR OPTION NUMBER 3.
C
C -27 THE RECIPROCAL OF THE BLOW-UP FACTOR FOR REJECTING VARIABLES
C MUST BE POSITIVE. NOW = ....
C
C -28 THE MAXIMUM NUMBER OF ITERATIONS = ... MUST BE POSITIVE.
C
C -29 THE OFFSET = ... BEYOND POSITION NCOLS = ... MUST BE POSTIVE
C FOR OPTION NUMBER 5.
C
C -30 THE FACTOR (NCOLS/MINPUT) WHERE PRETRIANGULARIZING IS
C PERFORMED MUST BE NONNEGATIVE. NOW = ....
C
C -31 THE NUMBER OF ROWS = ... MUST BE POSITIVE.
C
C -32 THE NUMBER OF COLUMNS = ... MUST BE POSTIVE.
C
C -33 THE ROW DIMENSION OF W(,) = ... MUST BE .GE. THE NUMBER OF
C ROWS = ....
C
C -34 FOR J = ... THE CONSTRAINT INDICATOR MUST BE 1-4.
C
C -35 FOR J = ... THE LOWER BOUND = ... IS .GT. THE UPPER BOUND =
C ....
C
C -36 THE INPUT ORDER OF COLUMNS = ... IS NOT BETWEEN 1 AND NCOLS
C = ....
C
C -37 THE BOUND POLARITY FLAG IN COMPONENT J = ... MUST BE
C POSITIVE. NOW = ....
C
C -38 THE ROW SEPARATOR TO APPLY WEIGHTING (...) MUST LIE BETWEEN
C 0 AND MINPUT = .... WEIGHT = ... MUST BE POSITIVE.
C
C***SEE ALSO DBOCLS, DBOLS
C***ROUTINES CALLED D1MACH, DAXPY, DCOPY, DDOT, DMOUT, DNRM2, DROT,
C DROTG, DSWAP, DVOUT, IVOUT, XERMSG
C***REVISION HISTORY (YYMMDD)
C 821220 DATE WRITTEN
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 900328 Added TYPE section. (WRB)
C 900510 Convert XERRWV calls to XERMSG calls. (RWC)
C 920422 Fixed usage of MINPUT. (WRB)
C 901009 Editorial changes, code now reads from top to bottom. (RWC)
C***END PROLOGUE DBOLSM
C
C PURPOSE
C -------
C THIS IS THE MAIN SUBPROGRAM THAT SOLVES THE BOUNDED
C LEAST SQUARES PROBLEM. THE PROBLEM SOLVED HERE IS:
C
C SOLVE E*X = F (LEAST SQUARES SENSE)
C WITH BOUNDS ON SELECTED X VALUES.
C
C TO CHANGE THIS SUBPROGRAM FROM SINGLE TO DOUBLE PRECISION BEGIN
C EDITING AT THE CARD 'C++'.
C CHANGE THE SUBPROGRAM NAME TO DBOLSM AND THE STRINGS
C /SAXPY/ TO /DAXPY/, /SCOPY/ TO /DCOPY/,
C /SDOT/ TO /DDOT/, /SNRM2/ TO /DNRM2/,
C /SROT/ TO /DROT/, /SROTG/ TO /DROTG/, /R1MACH/ TO /D1MACH/,
C /SVOUT/ TO /DVOUT/, /SMOUT/ TO /DMOUT/,
C /SSWAP/ TO /DSWAP/, /E0/ TO /D0/,
C /REAL / TO /DOUBLE PRECISION/.
C++
C
DOUBLE PRECISION W(MDW,*),BL(*),BU(*)
DOUBLE PRECISION X(*),RW(*),WW(*),SCL(*)
DOUBLE PRECISION ALPHA,BETA,BOU,COLABV,COLBLO
DOUBLE PRECISION CL1,CL2,CL3,ONE,BIG
DOUBLE PRECISION FAC,RNORM,SC,SS,T,TOLIND,WT
DOUBLE PRECISION TWO,T1,T2,WBIG,WLARGE,WMAG,XNEW
DOUBLE PRECISION ZERO,DDOT,DNRM2
DOUBLE PRECISION D1MACH,TOLSZE
INTEGER IBASIS(*),IBB(*),IND(*),IOPT(*)
LOGICAL FOUND,CONSTR
CHARACTER*8 XERN1, XERN2
CHARACTER*16 XERN3, XERN4
C
PARAMETER (ZERO=0.0D0, ONE=1.0D0, TWO=2.0D0)
C
INEXT(IDUM) = MIN(IDUM+1,MROWS)
C***FIRST EXECUTABLE STATEMENT DBOLSM
C
C Verify that the problem dimensions are defined properly.
C
IF (MINPUT.LE.0) THEN
WRITE (XERN1, '(I8)') MINPUT
CALL XERMSG ('SLATEC', 'DBOLSM', 'THE NUMBER OF ROWS = ' //
* XERN1 // ' MUST BE POSITIVE.', 31, 1)
MODE = -31
RETURN
ENDIF
C
IF (NCOLS.LE.0) THEN
WRITE (XERN1, '(I8)') NCOLS
CALL XERMSG ('SLATEC', 'DBOLSM', 'THE NUMBER OF COLUMNS = ' //
* XERN1 // ' MUST BE POSITIVE.', 32, 1)
MODE = -32
RETURN
ENDIF
C
IF (MDW.LT.MINPUT) THEN
WRITE (XERN1, '(I8)') MDW
WRITE (XERN2, '(I8)') MINPUT
CALL XERMSG ('SLATEC', 'DBOLSM',
* 'THE ROW DIMENSION OF W(,) = ' // XERN1 //
* ' MUST BE .GE. THE NUMBER OF ROWS = ' // XERN2, 33, 1)
MODE = -33
RETURN
ENDIF
C
C Verify that bound information is correct.
C
DO 10 J = 1,NCOLS
IF (IND(J).LT.1 .OR. IND(J).GT.4) THEN
WRITE (XERN1, '(I8)') J
WRITE (XERN2, '(I8)') IND(J)
CALL XERMSG ('SLATEC', 'DBOLSM', 'FOR J = ' // XERN1 //
* ' THE CONSTRAINT INDICATOR MUST BE 1-4', 34, 1)
MODE = -34
RETURN
ENDIF
10 CONTINUE
C
DO 20 J = 1,NCOLS
IF (IND(J).EQ.3) THEN
IF (BU(J).LT.BL(J)) THEN
WRITE (XERN1, '(I8)') J
WRITE (XERN3, '(1PD15.6)') BL(J)
WRITE (XERN4, '(1PD15.6)') BU(J)
CALL XERMSG ('SLATEC', 'DBOLSM', 'FOR J = ' // XERN1
* // ' THE LOWER BOUND = ' // XERN3 //
* ' IS .GT. THE UPPER BOUND = ' // XERN4, 35, 1)
MODE = -35
RETURN
ENDIF
ENDIF
20 CONTINUE
C
C Check that permutation and polarity arrays have been set.
C
DO 30 J = 1,NCOLS
IF (IBASIS(J).LT.1 .OR. IBASIS(J).GT.NCOLS) THEN
WRITE (XERN1, '(I8)') IBASIS(J)
WRITE (XERN2, '(I8)') NCOLS
CALL XERMSG ('SLATEC', 'DBOLSM',
* 'THE INPUT ORDER OF COLUMNS = ' // XERN1 //
* ' IS NOT BETWEEN 1 AND NCOLS = ' // XERN2, 36, 1)
MODE = -36
RETURN
ENDIF
C
IF (IBB(J).LE.0) THEN
WRITE (XERN1, '(I8)') J
WRITE (XERN2, '(I8)') IBB(J)
CALL XERMSG ('SLATEC', 'DBOLSM',
* 'THE BOUND POLARITY FLAG IN COMPONENT J = ' // XERN1 //
* ' MUST BE POSITIVE.$$NOW = ' // XERN2, 37, 1)
MODE = -37
RETURN
ENDIF
30 CONTINUE
C
C Process the option array.
C
FAC = 0.75D0
TOLIND = SQRT(D1MACH(4))
TOLSZE = SQRT(D1MACH(4))
ITMAX = 5*MAX(MINPUT,NCOLS)
WT = ONE
MVAL = 0
IPRINT = 0
C
C Changes to some parameters can occur through the option array,
C IOPT(*). Process this array looking carefully for input data
C errors.
C
LP = 0
LDS = 0
C
C Test for no more options.
C
590 LP = LP + LDS
IP = IOPT(LP+1)
JP = ABS(IP)
IF (IP.EQ.99) THEN
GO TO 470
ELSE IF (JP.EQ.99) THEN
LDS = 1
ELSE IF (JP.EQ.1) THEN
C
C Move the IOPT(*) processing pointer.
C
IF (IP.GT.0) THEN
LP = IOPT(LP+2) - 1
LDS = 0
ELSE
LDS = 2
ENDIF
ELSE IF (JP.EQ.2) THEN
C
C Change tolerance for rank determination.
C
IF (IP.GT.0) THEN
IOFF = IOPT(LP+2)
IF (IOFF.LE.0) THEN
WRITE (XERN1, '(I8)') IOFF
WRITE (XERN2, '(I8)') NCOLS
CALL XERMSG ('SLATEC', 'DBOLSM', 'THE OFFSET = ' //
* XERN1 // ' BEYOND POSITION NCOLS = ' // XERN2 //
* ' MUST BE POSITIVE FOR OPTION NUMBER 2.', 24, 1)
MODE = -24
RETURN
ENDIF
C
TOLIND = X(NCOLS+IOFF)
IF (TOLIND.LT.D1MACH(4)) THEN
WRITE (XERN3, '(1PD15.6)') TOLIND
WRITE (XERN4, '(1PD15.6)') D1MACH(4)
CALL XERMSG ('SLATEC', 'DBOLSM',
* 'THE TOLERANCE FOR RANK DETERMINATION = ' // XERN3
* // ' IS LESS THAN MACHINE PRECISION = ' // XERN4,
* 25, 0)
MODE = -25
ENDIF
ENDIF
LDS = 2
ELSE IF (JP.EQ.3) THEN
C
C Change blowup factor for allowing variables to become
C inactive.
C
IF (IP.GT.0) THEN
IOFF = IOPT(LP+2)
IF (IOFF.LE.0) THEN
WRITE (XERN1, '(I8)') IOFF
WRITE (XERN2, '(I8)') NCOLS
CALL XERMSG ('SLATEC', 'DBOLSM', 'THE OFFSET = ' //
* XERN1 // ' BEYOND POSITION NCOLS = ' // XERN2 //
* ' MUST BE POSITIVE FOR OPTION NUMBER 3.', 26, 1)
MODE = -26
RETURN
ENDIF
C
TOLSZE = X(NCOLS+IOFF)
IF (TOLSZE.LE.ZERO) THEN
WRITE (XERN3, '(1PD15.6)') TOLSZE
CALL XERMSG ('SLATEC', 'DBOLSM', 'THE RECIPROCAL ' //
* 'OF THE BLOW-UP FACTOR FOR REJECTING VARIABLES ' //
* 'MUST BE POSITIVE.$$NOW = ' // XERN3, 27, 1)
MODE = -27
RETURN
ENDIF
ENDIF
LDS = 2
ELSE IF (JP.EQ.4) THEN
C
C Change the maximum number of iterations allowed.
C
IF (IP.GT.0) THEN
ITMAX = IOPT(LP+2)
IF (ITMAX.LE.0) THEN
WRITE (XERN1, '(I8)') ITMAX
CALL XERMSG ('SLATEC', 'DBOLSM',
* 'THE MAXIMUM NUMBER OF ITERATIONS = ' // XERN1 //
* ' MUST BE POSITIVE.', 28, 1)
MODE = -28
RETURN
ENDIF
ENDIF
LDS = 2
ELSE IF (JP.EQ.5) THEN
C
C Change the factor for pretriangularizing the data matrix.
C
IF (IP.GT.0) THEN
IOFF = IOPT(LP+2)
IF (IOFF.LE.0) THEN
WRITE (XERN1, '(I8)') IOFF
WRITE (XERN2, '(I8)') NCOLS
CALL XERMSG ('SLATEC', 'DBOLSM', 'THE OFFSET = ' //
* XERN1 // ' BEYOND POSITION NCOLS = ' // XERN2 //
* ' MUST BE POSITIVE FOR OPTION NUMBER 5.', 29, 1)
MODE = -29
RETURN
ENDIF
C
FAC = X(NCOLS+IOFF)
IF (FAC.LT.ZERO) THEN
WRITE (XERN3, '(1PD15.6)') FAC
CALL XERMSG ('SLATEC', 'DBOLSM',
* 'THE FACTOR (NCOLS/MINPUT) WHERE PRE-' //
* 'TRIANGULARIZING IS PERFORMED MUST BE NON-' //
* 'NEGATIVE.$$NOW = ' // XERN3, 30, 0)
MODE = -30
RETURN
ENDIF
ENDIF
LDS = 2
ELSE IF (JP.EQ.6) THEN
C
C Change the weighting factor (from 1.0) to apply to components
C numbered .gt. MVAL (initially set to 1.) This trick is needed
C for applications of this subprogram to the heavily weighted
C least squares problem that come from equality constraints.
C
IF (IP.GT.0) THEN
IOFF = IOPT(LP+2)
MVAL = IOPT(LP+3)
WT = X(NCOLS+IOFF)
ENDIF
C
IF (MVAL.LT.0 .OR. MVAL.GT.MINPUT .OR. WT.LE.ZERO) THEN
WRITE (XERN1, '(I8)') MVAL
WRITE (XERN2, '(I8)') MINPUT
WRITE (XERN3, '(1PD15.6)') WT
CALL XERMSG ('SLATEC', 'DBOLSM',
* 'THE ROW SEPARATOR TO APPLY WEIGHTING (' // XERN1 //
* ') MUST LIE BETWEEN 0 AND MINPUT = ' // XERN2 //
* '.$$WEIGHT = ' // XERN3 // ' MUST BE POSITIVE.', 38, 0)
MODE = -38
RETURN
ENDIF
LDS = 3
ELSE IF (JP.EQ.7) THEN
C
C Turn on debug output.
C
IF (IP.GT.0) IPRINT = 1
LDS = 2
ELSE
WRITE (XERN1, '(I8)') IP
CALL XERMSG ('SLATEC', 'DBOLSM', 'THE OPTION NUMBER = ' //
* XERN1 // ' IS NOT DEFINED.', 23, 1)
MODE = -23
RETURN
ENDIF
GO TO 590
C
C Pretriangularize rectangular arrays of certain sizes for
C increased efficiency.
C
470 IF (FAC*MINPUT.GT.NCOLS) THEN
DO 490 J = 1,NCOLS+1
DO 480 I = MINPUT,J+MVAL+1,-1
CALL DROTG(W(I-1,J),W(I,J),SC,SS)
W(I,J) = ZERO
CALL DROT(NCOLS-J+1,W(I-1,J+1),MDW,W(I,J+1),MDW,SC,SS)
480 CONTINUE
490 CONTINUE
MROWS = NCOLS + MVAL + 1
ELSE
MROWS = MINPUT
ENDIF
C
C Set the X(*) array to zero so all components are defined.
C
CALL DCOPY(NCOLS,ZERO,0,X,1)
C
C The arrays IBASIS(*) and IBB(*) are initialized by the calling
C program and the column scaling is defined in the calling program.
C 'BIG' is plus infinity on this machine.
C
BIG = D1MACH(2)
DO 550 J = 1,NCOLS
IF (IND(J).EQ.1) THEN
BU(J) = BIG
ELSE IF (IND(J).EQ.2) THEN
BL(J) = -BIG
ELSE IF (IND(J).EQ.4) THEN
BL(J) = -BIG
BU(J) = BIG
ENDIF
550 CONTINUE
C
DO 570 J = 1,NCOLS
IF ((BL(J).LE.ZERO.AND.ZERO.LE.BU(J).AND.ABS(BU(J)).LT.
* ABS(BL(J))) .OR. BU(J).LT.ZERO) THEN
T = BU(J)
BU(J) = -BL(J)
BL(J) = -T
SCL(J) = -SCL(J)
DO 560 I = 1,MROWS
W(I,J) = -W(I,J)
560 CONTINUE
ENDIF
C
C Indices in set T(=TIGHT) are denoted by negative values
C of IBASIS(*).
C
IF (BL(J).GE.ZERO) THEN
IBASIS(J) = -IBASIS(J)
T = -BL(J)
BU(J) = BU(J) + T
CALL DAXPY(MROWS,T,W(1,J),1,W(1,NCOLS+1),1)
ENDIF
570 CONTINUE
C
NSETB = 0
ITER = 0
C
IF (IPRINT.GT.0) THEN
CALL DMOUT(MROWS,NCOLS+1,MDW,W,'('' PRETRI. INPUT MATRIX'')',
* -4)
CALL DVOUT(NCOLS,BL,'('' LOWER BOUNDS'')',-4)
CALL DVOUT(NCOLS,BU,'('' UPPER BOUNDS'')',-4)
ENDIF
C
580 ITER = ITER + 1
IF (ITER.GT.ITMAX) THEN
WRITE (XERN1, '(I8)') ITMAX
CALL XERMSG ('SLATEC', 'DBOLSM', 'MORE THAN ITMAX = ' // XERN1
* // ' ITERATIONS SOLVING BOUNDED LEAST SQUARES PROBLEM.',
* 22, 1)
MODE = -22
C
C Rescale and translate variables.
C
IGOPR = 1
GO TO 130
ENDIF
C
C Find a variable to become non-active.
C T
C Compute (negative) of gradient vector, W = E *(F-E*X).
C
CALL DCOPY(NCOLS,ZERO,0,WW,1)
DO 200 J = NSETB+1,NCOLS
JCOL = ABS(IBASIS(J))
WW(J) = DDOT(MROWS-NSETB,W(INEXT(NSETB),J),1,
* W(INEXT(NSETB),NCOLS+1),1)*ABS(SCL(JCOL))
200 CONTINUE
C
IF (IPRINT.GT.0) THEN
CALL DVOUT(NCOLS,WW,'('' GRADIENT VALUES'')',-4)
CALL IVOUT(NCOLS,IBASIS,'('' INTERNAL VARIABLE ORDER'')',-4)
CALL IVOUT(NCOLS,IBB,'('' BOUND POLARITY'')',-4)
ENDIF
C
C If active set = number of total rows, quit.
C
210 IF (NSETB.EQ.MROWS) THEN
FOUND = .FALSE.
GO TO 120
ENDIF
C
C Choose an extremal component of gradient vector for a candidate
C to become non-active.
C
WLARGE = -BIG
WMAG = -BIG
DO 220 J = NSETB+1,NCOLS
T = WW(J)
IF (T.EQ.BIG) GO TO 220
ITEMP = IBASIS(J)
JCOL = ABS(ITEMP)
T1 = DNRM2(MVAL-NSETB,W(INEXT(NSETB),J),1)
IF (ITEMP.LT.0) THEN
IF (MOD(IBB(JCOL),2).EQ.0) T = -T
IF (T.LT.ZERO) GO TO 220
IF (MVAL.GT.NSETB) T = T1
IF (T.GT.WLARGE) THEN
WLARGE = T
JLARGE = J
ENDIF
ELSE
IF (MVAL.GT.NSETB) T = T1
IF (ABS(T).GT.WMAG) THEN
WMAG = ABS(T)
JMAG = J
ENDIF
ENDIF
220 CONTINUE
C
C Choose magnitude of largest component of gradient for candidate.
C
JBIG = 0
WBIG = ZERO
IF (WLARGE.GT.ZERO) THEN
JBIG = JLARGE
WBIG = WLARGE
ENDIF
C
IF (WMAG.GE.WBIG) THEN
JBIG = JMAG
WBIG = WMAG
ENDIF
C
IF (JBIG.EQ.0) THEN
FOUND = .FALSE.
IF (IPRINT.GT.0) THEN
CALL IVOUT(0,I,'('' FOUND NO VARIABLE TO ENTER'')',-4)
ENDIF
GO TO 120
ENDIF
C
C See if the incoming column is sufficiently independent. This
C test is made before an elimination is performed.
C
IF (IPRINT.GT.0)
* CALL IVOUT(1,JBIG,'('' TRY TO BRING IN THIS COL.'')',-4)
C
IF (MVAL.LE.NSETB) THEN
CL1 = DNRM2(MVAL,W(1,JBIG),1)
CL2 = ABS(WT)*DNRM2(NSETB-MVAL,W(INEXT(MVAL),JBIG),1)
CL3 = ABS(WT)*DNRM2(MROWS-NSETB,W(INEXT(NSETB),JBIG),1)
CALL DROTG(CL1,CL2,SC,SS)
COLABV = ABS(CL1)
COLBLO = CL3
ELSE
CL1 = DNRM2(NSETB,W(1,JBIG),1)
CL2 = DNRM2(MVAL-NSETB,W(INEXT(NSETB),JBIG),1)
CL3 = ABS(WT)*DNRM2(MROWS-MVAL,W(INEXT(MVAL),JBIG),1)
COLABV = CL1
CALL DROTG(CL2,CL3,SC,SS)
COLBLO = ABS(CL2)
ENDIF
C
IF (COLBLO.LE.TOLIND*COLABV) THEN
WW(JBIG) = BIG
IF (IPRINT.GT.0)
* CALL IVOUT(0,I,'('' VARIABLE IS DEPENDENT, NOT USED.'')',
* -4)
GO TO 210
ENDIF
C
C Swap matrix columns NSETB+1 and JBIG, plus pointer information,
C and gradient values.
C
NSETB = NSETB + 1
IF (NSETB.NE.JBIG) THEN
CALL DSWAP(MROWS,W(1,NSETB),1,W(1,JBIG),1)
CALL DSWAP(1,WW(NSETB),1,WW(JBIG),1)
ITEMP = IBASIS(NSETB)
IBASIS(NSETB) = IBASIS(JBIG)
IBASIS(JBIG) = ITEMP
ENDIF
C
C Eliminate entries below the pivot line in column NSETB.
C
IF (MROWS.GT.NSETB) THEN
DO 230 I = MROWS,NSETB+1,-1
IF (I.EQ.MVAL+1) GO TO 230
CALL DROTG(W(I-1,NSETB),W(I,NSETB),SC,SS)
W(I,NSETB) = ZERO
CALL DROT(NCOLS-NSETB+1,W(I-1,NSETB+1),MDW,W(I,NSETB+1),
* MDW,SC,SS)
230 CONTINUE
C
IF (MVAL.GE.NSETB .AND. MVAL.LT.MROWS) THEN
CALL DROTG(W(NSETB,NSETB),W(MVAL+1,NSETB),SC,SS)
W(MVAL+1,NSETB) = ZERO
CALL DROT(NCOLS-NSETB+1,W(NSETB,NSETB+1),MDW,
* W(MVAL+1,NSETB+1),MDW,SC,SS)
ENDIF
ENDIF
C
IF (W(NSETB,NSETB).EQ.ZERO) THEN
WW(NSETB) = BIG
NSETB = NSETB - 1
IF (IPRINT.GT.0) THEN
CALL IVOUT(0,I,'('' PIVOT IS ZERO, NOT USED.'')',-4)
ENDIF
GO TO 210
ENDIF
C
C Check that new variable is moving in the right direction.
C
ITEMP = IBASIS(NSETB)
JCOL = ABS(ITEMP)
XNEW = (W(NSETB,NCOLS+1)/W(NSETB,NSETB))/ABS(SCL(JCOL))
IF (ITEMP.LT.0) THEN
C
C IF(WW(NSETB).GE.ZERO.AND.XNEW.LE.ZERO) exit(quit)
C IF(WW(NSETB).LE.ZERO.AND.XNEW.GE.ZERO) exit(quit)
C
IF ((WW(NSETB).GE.ZERO.AND.XNEW.LE.ZERO) .OR.
* (WW(NSETB).LE.ZERO.AND.XNEW.GE.ZERO)) GO TO 240
ENDIF
FOUND = .TRUE.
GO TO 120
C
240 WW(NSETB) = BIG
NSETB = NSETB - 1
IF (IPRINT.GT.0)
* CALL IVOUT(0,I,'('' VARIABLE HAS BAD DIRECTION, NOT USED.'')',
* -4)
GO TO 210
C
C Solve the triangular system.
C
270 CALL DCOPY(NSETB,W(1,NCOLS+1),1,RW,1)
DO 280 J = NSETB,1,-1
RW(J) = RW(J)/W(J,J)
JCOL = ABS(IBASIS(J))
T = RW(J)
IF (MOD(IBB(JCOL),2).EQ.0) RW(J) = -RW(J)
CALL DAXPY(J-1,-T,W(1,J),1,RW,1)
RW(J) = RW(J)/ABS(SCL(JCOL))
280 CONTINUE
C
IF (IPRINT.GT.0) THEN
CALL DVOUT(NSETB,RW,'('' SOLN. VALUES'')',-4)
CALL IVOUT(NSETB,IBASIS,'('' COLS. USED'')',-4)
ENDIF
C
IF (LGOPR.EQ.2) THEN
CALL DCOPY(NSETB,RW,1,X,1)
DO 450 J = 1,NSETB
ITEMP = IBASIS(J)
JCOL = ABS(ITEMP)
IF (ITEMP.LT.0) THEN
BOU = ZERO
ELSE
BOU = BL(JCOL)
ENDIF
C
IF ((-BOU).NE.BIG) BOU = BOU/ABS(SCL(JCOL))
IF (X(J).LE.BOU) THEN
JDROP1 = J
GO TO 340
ENDIF
C
BOU = BU(JCOL)
IF (BOU.NE.BIG) BOU = BOU/ABS(SCL(JCOL))
IF (X(J).GE.BOU) THEN
JDROP2 = J
GO TO 340
ENDIF
450 CONTINUE
GO TO 340
ENDIF
C
C See if the unconstrained solution (obtained by solving the
C triangular system) satisfies the problem bounds.
C
ALPHA = TWO
BETA = TWO
X(NSETB) = ZERO
DO 310 J = 1,NSETB
ITEMP = IBASIS(J)
JCOL = ABS(ITEMP)
T1 = TWO
T2 = TWO
IF (ITEMP.LT.0) THEN
BOU = ZERO
ELSE
BOU = BL(JCOL)
ENDIF
IF ((-BOU).NE.BIG) BOU = BOU/ABS(SCL(JCOL))
IF (RW(J).LE.BOU) T1 = (X(J)-BOU)/ (X(J)-RW(J))
BOU = BU(JCOL)
IF (BOU.NE.BIG) BOU = BOU/ABS(SCL(JCOL))
IF (RW(J).GE.BOU) T2 = (BOU-X(J))/ (RW(J)-X(J))
C
C If not, then compute a step length so that the variables remain
C feasible.
C
IF (T1.LT.ALPHA) THEN
ALPHA = T1
JDROP1 = J
ENDIF
C
IF (T2.LT.BETA) THEN
BETA = T2
JDROP2 = J
ENDIF
310 CONTINUE
C
CONSTR = ALPHA .LT. TWO .OR. BETA .LT. TWO
IF (.NOT.CONSTR) THEN
C
C Accept the candidate because it satisfies the stated bounds
C on the variables.
C
CALL DCOPY(NSETB,RW,1,X,1)
GO TO 580
ENDIF
C
C Take a step that is as large as possible with all variables
C remaining feasible.
C
DO 330 J = 1,NSETB
X(J) = X(J) + MIN(ALPHA,BETA)* (RW(J)-X(J))
330 CONTINUE
C
IF (ALPHA.LE.BETA) THEN
JDROP2 = 0
ELSE
JDROP1 = 0
ENDIF
C
340 IF (JDROP1+JDROP2.LE.0 .OR. NSETB.LE.0) GO TO 580
350 JDROP = JDROP1 + JDROP2
ITEMP = IBASIS(JDROP)
JCOL = ABS(ITEMP)
IF (JDROP2.GT.0) THEN
C
C Variable is at an upper bound. Subtract multiple of this
C column from right hand side.
C
T = BU(JCOL)
IF (ITEMP.GT.0) THEN
BU(JCOL) = T - BL(JCOL)
BL(JCOL) = -T
ITEMP = -ITEMP
SCL(JCOL) = -SCL(JCOL)
DO 360 I = 1,JDROP
W(I,JDROP) = -W(I,JDROP)
360 CONTINUE
ELSE
IBB(JCOL) = IBB(JCOL) + 1
IF (MOD(IBB(JCOL),2).EQ.0) T = -T
ENDIF
C
C Variable is at a lower bound.
C
ELSE
IF (ITEMP.LT.ZERO) THEN
T = ZERO
ELSE
T = -BL(JCOL)
BU(JCOL) = BU(JCOL) + T
ITEMP = -ITEMP
ENDIF
ENDIF
C
CALL DAXPY(JDROP,T,W(1,JDROP),1,W(1,NCOLS+1),1)
C
C Move certain columns left to achieve upper Hessenberg form.
C
CALL DCOPY(JDROP,W(1,JDROP),1,RW,1)
DO 370 J = JDROP+1,NSETB
IBASIS(J-1) = IBASIS(J)
X(J-1) = X(J)
CALL DCOPY(J,W(1,J),1,W(1,J-1),1)
370 CONTINUE
C
IBASIS(NSETB) = ITEMP
W(1,NSETB) = ZERO
CALL DCOPY(MROWS-JDROP,W(1,NSETB),0,W(JDROP+1,NSETB),1)
CALL DCOPY(JDROP,RW,1,W(1,NSETB),1)
C
C Transform the matrix from upper Hessenberg form to upper
C triangular form.
C
NSETB = NSETB - 1
DO 390 I = JDROP,NSETB
C
C Look for small pivots and avoid mixing weighted and
C nonweighted rows.
C
IF (I.EQ.MVAL) THEN
T = ZERO
DO 380 J = I,NSETB
JCOL = ABS(IBASIS(J))
T1 = ABS(W(I,J)*SCL(JCOL))
IF (T1.GT.T) THEN
JBIG = J
T = T1
ENDIF
380 CONTINUE
GO TO 400
ENDIF
CALL DROTG(W(I,I),W(I+1,I),SC,SS)
W(I+1,I) = ZERO
CALL DROT(NCOLS-I+1,W(I,I+1),MDW,W(I+1,I+1),MDW,SC,SS)
390 CONTINUE
GO TO 430
C
C The triangularization is completed by giving up the Hessenberg
C form and triangularizing a rectangular matrix.
C
400 CALL DSWAP(MROWS,W(1,I),1,W(1,JBIG),1)
CALL DSWAP(1,WW(I),1,WW(JBIG),1)
CALL DSWAP(1,X(I),1,X(JBIG),1)
ITEMP = IBASIS(I)
IBASIS(I) = IBASIS(JBIG)
IBASIS(JBIG) = ITEMP
JBIG = I
DO 420 J = JBIG,NSETB
DO 410 I = J+1,MROWS
CALL DROTG(W(J,J),W(I,J),SC,SS)
W(I,J) = ZERO
CALL DROT(NCOLS-J+1,W(J,J+1),MDW,W(I,J+1),MDW,SC,SS)
410 CONTINUE
420 CONTINUE
C
C See if the remaining coefficients are feasible. They should be
C because of the way MIN(ALPHA,BETA) was chosen. Any that are not
C feasible will be set to their bounds and appropriately translated.
C
430 JDROP1 = 0
JDROP2 = 0
LGOPR = 2
GO TO 270
C
C Find a variable to become non-active.
C
120 IF (FOUND) THEN
LGOPR = 1
GO TO 270
ENDIF
C
C Rescale and translate variables.
C
IGOPR = 2
130 CALL DCOPY(NSETB,X,1,RW,1)
CALL DCOPY(NCOLS,ZERO,0,X,1)
DO 140 J = 1,NSETB
JCOL = ABS(IBASIS(J))
X(JCOL) = RW(J)*ABS(SCL(JCOL))
140 CONTINUE
C
DO 150 J = 1,NCOLS
IF (MOD(IBB(J),2).EQ.0) X(J) = BU(J) - X(J)
150 CONTINUE
C
DO 160 J = 1,NCOLS
JCOL = IBASIS(J)
IF (JCOL.LT.0) X(-JCOL) = BL(-JCOL) + X(-JCOL)
160 CONTINUE
C
DO 170 J = 1,NCOLS
IF (SCL(J).LT.ZERO) X(J) = -X(J)
170 CONTINUE
C
I = MAX(NSETB,MVAL)
RNORM = DNRM2(MROWS-I,W(INEXT(I),NCOLS+1),1)
C
IF (IGOPR.EQ.2) MODE = NSETB
RETURN
END
