*DECK BNDSOL
SUBROUTINE BNDSOL (MODE, G, MDG, NB, IP, IR, X, N, RNORM)
C***BEGIN PROLOGUE BNDSOL
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 SINGLE 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 BNDACC 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 BNDSOL(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 BNDACC 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 BNDACC(...) Introduce new blocks of data.
C
C CALL BNDSOL(1,...)Compute solution vector and length of
C residual vector.
C
C CALL BNDSOL(2,...)Given any row vector H solve YR = H for the
C row vector Y.
C
C CALL BNDSOL(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 BNDACC..
C
C The entire set of parameters for BNDACC are
C
C Input..
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 BNDACC. Its subsequent value
C is controlled by BNDACC to set up for the
C next call to BNDACC.
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 BNDACC. Its subsequent value is controlled
C by BNDACC. 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..
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 BNDACC.
C
C IP,IR The values of these arguments are advanced by
C BNDACC to be ready for storing and processing
C a new block of data in G(*,*).
C
C Description of calling sequence for BNDSOL..
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 BNDSOL 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 BNDACC.
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 BNDSOL 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 BNDACC(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 900326 Removed duplicate information from DESCRIPTION section.
C (WRB)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE BNDSOL
DIMENSION G(MDG,*),X(*)
C***FIRST EXECUTABLE STATEMENT BNDSOL
ZERO=0.
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', 'BNDSOL',
+ 'A ZERO DIAGONAL TERM IS IN THE N BY N UPPER TRIANGULAR ' //
+ 'MATRIX.', NERR, IOPT)
RETURN
END