*DECK DBNDSL SUBROUTINE DBNDSL (MODE, G, MDG, NB, IP, IR, X, N, RNORM) C***BEGIN PROLOGUE DBNDSL C***PURPOSE Solve the least squares problem for a banded matrix using C sequential accumulation of rows of the data matrix. C Exactly one right-hand side vector is permitted. C***LIBRARY SLATEC C***CATEGORY D9 C***TYPE DOUBLE PRECISION (BNDSOL-S, DBNDSL-D) C***KEYWORDS BANDED MATRIX, CURVE FITTING, LEAST SQUARES C***AUTHOR Lawson, C. L., (JPL) C Hanson, R. J., (SNLA) C***DESCRIPTION C C These subroutines solve the least squares problem Ax = b for C banded matrices A using sequential accumulation of rows of the C data matrix. Exactly one right-hand side vector is permitted. C C These subroutines are intended for the type of least squares C systems that arise in applications such as curve or surface C fitting of data. The least squares equations are accumulated and C processed using only part of the data. This requires a certain C user interaction during the solution of Ax = b. C C Specifically, suppose the data matrix (A B) is row partitioned C into Q submatrices. Let (E F) be the T-th one of these C submatrices where E = (0 C 0). Here the dimension of E is MT by N C and the dimension of C is MT by NB. The value of NB is the C bandwidth of A. The dimensions of the leading block of zeros in E C are MT by JT-1. C C The user of the subroutine DBNDAC provides MT,JT,C and F for C T=1,...,Q. Not all of this data must be supplied at once. C C Following the processing of the various blocks (E F), the matrix C (A B) has been transformed to the form (R D) where R is upper C triangular and banded with bandwidth NB. The least squares C system Rx = d is then easily solved using back substitution by C executing the statement CALL DBNDSL(1,...). The sequence of C values for JT must be nondecreasing. This may require some C preliminary interchanges of rows and columns of the matrix A. C C The primary reason for these subroutines is that the total C processing can take place in a working array of dimension MU by C NB+1. An acceptable value for MU is C C MU = MAX(MT + N + 1), C C where N is the number of unknowns. C C Here the maximum is taken over all values of MT for T=1,...,Q. C Notice that MT can be taken to be a small as one, showing that C MU can be as small as N+2. The subprogram DBNDAC processes the C rows more efficiently if MU is large enough so that each new C block (C F) has a distinct value of JT. C C The four principle parts of these algorithms are obtained by the C following call statements C C CALL DBNDAC(...) Introduce new blocks of data. C C CALL DBNDSL(1,...)Compute solution vector and length of C residual vector. C C CALL DBNDSL(2,...)Given any row vector H solve YR = H for the C row vector Y. C C CALL DBNDSL(3,...)Given any column vector W solve RZ = W for C the column vector Z. C C The dots in the above call statements indicate additional C arguments that will be specified in the following paragraphs. C C The user must dimension the array appearing in the call list.. C G(MDG,NB+1) C C Description of calling sequence for DBNDAC.. C C The entire set of parameters for DBNDAC are C C Input.. All Type REAL variables are DOUBLE PRECISION C C G(*,*) The working array into which the user will C place the MT by NB+1 block (C F) in rows IR C through IR+MT-1, columns 1 through NB+1. C See descriptions of IR and MT below. C C MDG The number of rows in the working array C G(*,*). The value of MDG should be .GE. MU. C The value of MU is defined in the abstract C of these subprograms. C C NB The bandwidth of the data matrix A. C C IP Set by the user to the value 1 before the C first call to DBNDAC. Its subsequent value C is controlled by DBNDAC to set up for the C next call to DBNDAC. C C IR Index of the row of G(*,*) where the user is C the user to the value 1 before the first call C to DBNDAC. Its subsequent value is controlled C by DBNDAC. A value of IR .GT. MDG is considered C an error. C C MT,JT Set by the user to indicate respectively the C number of new rows of data in the block and C the index of the first nonzero column in that C set of rows (E F) = (0 C 0 F) being processed. C Output.. All Type REAL variables are DOUBLE PRECISION C C G(*,*) The working array which will contain the C processed rows of that part of the data C matrix which has been passed to DBNDAC. C C IP,IR The values of these arguments are advanced by C DBNDAC to be ready for storing and processing C a new block of data in G(*,*). C C Description of calling sequence for DBNDSL.. C C The user must dimension the arrays appearing in the call list.. C C G(MDG,NB+1), X(N) C C The entire set of parameters for DBNDSL are C C Input.. C C MODE Set by the user to one of the values 1, 2, or C 3. These values respectively indicate that C the solution of AX = B, YR = H or RZ = W is C required. C C G(*,*),MDG, These arguments all have the same meaning and C NB,IP,IR contents as following the last call to DBNDAC. C C X(*) With mode=2 or 3 this array contains, C respectively, the right-side vectors H or W of C the systems YR = H or RZ = W. C C N The number of variables in the solution C vector. If any of the N diagonal terms are C zero the subroutine DBNDSL prints an C appropriate message. This condition is C considered an error. C C Output.. C C X(*) This array contains the solution vectors X, C Y or Z of the systems AX = B, YR = H or C RZ = W depending on the value of MODE=1, C 2 or 3. C C RNORM If MODE=1 RNORM is the Euclidean length of the C residual vector AX-B. When MODE=2 or 3 RNORM C is set to zero. C C Remarks.. C C To obtain the upper triangular matrix and transformed right-hand C side vector D so that the super diagonals of R form the columns C of G(*,*), execute the following Fortran statements. C C NBP1=NB+1 C C DO 10 J=1, NBP1 C C 10 G(IR,J) = 0.E0 C C MT=1 C C JT=N+1 C C CALL DBNDAC(G,MDG,NB,IP,IR,MT,JT) C C***REFERENCES C. L. Lawson and R. J. Hanson, Solving Least Squares C Problems, Prentice-Hall, Inc., 1974, Chapter 27. C***ROUTINES CALLED XERMSG C***REVISION HISTORY (YYMMDD) C 790101 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890831 Modified array declarations. (WRB) C 891006 Cosmetic changes to prologue. (WRB) 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 920501 Reformatted the REFERENCES section. (WRB) C***END PROLOGUE DBNDSL IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION G(MDG,*),X(*) C***FIRST EXECUTABLE STATEMENT DBNDSL ZERO=0.D0 C RNORM=ZERO GO TO (10,90,50), MODE C ********************* MODE = 1 C ALG. STEP 26 10 DO 20 J=1,N X(J)=G(J,NB+1) 20 CONTINUE RSQ=ZERO NP1=N+1 IRM1=IR-1 IF (NP1.GT.IRM1) GO TO 40 DO 30 J=NP1,IRM1 RSQ=RSQ+G(J,NB+1)**2 30 CONTINUE RNORM=SQRT(RSQ) 40 CONTINUE C ********************* MODE = 3 C ALG. STEP 27 50 DO 80 II=1,N I=N+1-II C ALG. STEP 28 S=ZERO L=MAX(0,I-IP) C ALG. STEP 29 IF (I.EQ.N) GO TO 70 C ALG. STEP 30 IE=MIN(N+1-I,NB) DO 60 J=2,IE JG=J+L IX=I-1+J S=S+G(I,JG)*X(IX) 60 CONTINUE C ALG. STEP 31 70 IF (G(I,L+1)) 80,130,80 80 X(I)=(X(I)-S)/G(I,L+1) C ALG. STEP 32 RETURN C ********************* MODE = 2 90 DO 120 J=1,N S=ZERO IF (J.EQ.1) GO TO 110 I1=MAX(1,J-NB+1) I2=J-1 DO 100 I=I1,I2 L=J-I+1+MAX(0,I-IP) S=S+X(I)*G(I,L) 100 CONTINUE 110 L=MAX(0,J-IP) IF (G(J,L+1)) 120,130,120 120 X(J)=(X(J)-S)/G(J,L+1) RETURN C 130 CONTINUE NERR=1 IOPT=2 CALL XERMSG ('SLATEC', 'DBNDSL', + 'A ZERO DIAGONAL TERM IS IN THE N BY N UPPER TRIANGULAR ' // + 'MATRIX.', NERR, IOPT) RETURN END