*DECK DBNDAC
SUBROUTINE DBNDAC (G, MDG, NB, IP, IR, MT, JT)
C***BEGIN PROLOGUE DBNDAC
C***PURPOSE Compute the LU factorization of a banded matrices 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 (BNDACC-S, DBNDAC-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 to place the new block of data (C F). Set by
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
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.. All Type REAL variables are DOUBLE PRECISION
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.. All Type REAL variables are DOUBLE PRECISION
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 DH12, XERMSG
C***REVISION HISTORY (YYMMDD)
C 790101 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (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 DBNDAC
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
DIMENSION G(MDG,*)
C***FIRST EXECUTABLE STATEMENT DBNDAC
ZERO=0.D0
C
C ALG. STEPS 1-4 ARE PERFORMED EXTERNAL TO THIS SUBROUTINE.
C
NBP1=NB+1
IF (MT.LE.0.OR.NB.LE.0) RETURN
C
IF(.NOT.MDG.LT.IR) GO TO 5
NERR=1
IOPT=2
CALL XERMSG ('SLATEC', 'DBNDAC', 'MDG.LT.IR, PROBABLE ERROR.',
+ NERR, IOPT)
RETURN
5 CONTINUE
C
C ALG. STEP 5
IF (JT.EQ.IP) GO TO 70
C ALG. STEPS 6-7
IF (JT.LE.IR) GO TO 30
C ALG. STEPS 8-9
DO 10 I=1,MT
IG1=JT+MT-I
IG2=IR+MT-I
DO 10 J=1,NBP1
G(IG1,J)=G(IG2,J)
10 CONTINUE
C ALG. STEP 10
IE=JT-IR
DO 20 I=1,IE
IG=IR+I-1
DO 20 J=1,NBP1
G(IG,J)=ZERO
20 CONTINUE
C ALG. STEP 11
IR=JT
C ALG. STEP 12
30 MU=MIN(NB-1,IR-IP-1)
IF (MU.EQ.0) GO TO 60
C ALG. STEP 13
DO 50 L=1,MU
C ALG. STEP 14
K=MIN(L,JT-IP)
C ALG. STEP 15
LP1=L+1
IG=IP+L
DO 40 I=LP1,NB
JG=I-K
G(IG,JG)=G(IG,I)
40 CONTINUE
C ALG. STEP 16
DO 50 I=1,K
JG=NBP1-I
G(IG,JG)=ZERO
50 CONTINUE
C ALG. STEP 17
60 IP=JT
C ALG. STEPS 18-19
70 MH=IR+MT-IP
KH=MIN(NBP1,MH)
C ALG. STEP 20
DO 80 I=1,KH
CALL DH12 (1,I,MAX(I+1,IR-IP+1),MH,G(IP,I),1,RHO,
1 G(IP,I+1),1,MDG,NBP1-I)
80 CONTINUE
C ALG. STEP 21
IR=IP+KH
C ALG. STEP 22
IF (KH.LT.NBP1) GO TO 100
C ALG. STEP 23
DO 90 I=1,NB
G(IR-1,I)=ZERO
90 CONTINUE
C ALG. STEP 24
100 CONTINUE
C ALG. STEP 25
RETURN
END