*DECK BANDV
SUBROUTINE BANDV (NM, N, MBW, A, E21, M, W, Z, IERR, NV, RV, RV6)
C***BEGIN PROLOGUE BANDV
C***PURPOSE Form the eigenvectors of a real symmetric band matrix
C associated with a set of ordered approximate eigenvalues
C by inverse iteration.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4C3
C***TYPE SINGLE PRECISION (BANDV-S)
C***KEYWORDS EIGENVECTORS, EISPACK
C***AUTHOR Smith, B. T., et al.
C***DESCRIPTION
C
C This subroutine finds those eigenvectors of a REAL SYMMETRIC
C BAND matrix corresponding to specified eigenvalues, using inverse
C iteration. The subroutine may also be used to solve systems
C of linear equations with a symmetric or non-symmetric band
C coefficient matrix.
C
C On INPUT
C
C NM must be set to the row dimension of the two-dimensional
C array parameters, A and Z, as declared in the calling
C program dimension statement. NM is an INTEGER variable.
C
C N is the order of the matrix A. N is an INTEGER variable.
C N must be less than or equal to NM.
C
C MBW is the number of columns of the array A used to store the
C band matrix. If the matrix is symmetric, MBW is its (half)
C band width, denoted MB and defined as the number of adjacent
C diagonals, including the principal diagonal, required to
C specify the non-zero portion of the lower triangle of the
C matrix. If the subroutine is being used to solve systems
C of linear equations and the coefficient matrix is not
C symmetric, it must however have the same number of adjacent
C diagonals above the main diagonal as below, and in this
C case, MBW=2*MB-1. MBW is an INTEGER variable. MB must not
C be greater than N.
C
C A contains the lower triangle of the symmetric band input
C matrix stored as an N by MB array. Its lowest subdiagonal
C is stored in the last N+1-MB positions of the first column,
C its next subdiagonal in the last N+2-MB positions of the
C second column, further subdiagonals similarly, and finally
C its principal diagonal in the N positions of column MB.
C If the subroutine is being used to solve systems of linear
C equations and the coefficient matrix is not symmetric, A is
C N by 2*MB-1 instead with lower triangle as above and with
C its first superdiagonal stored in the first N-1 positions of
C column MB+1, its second superdiagonal in the first N-2
C positions of column MB+2, further superdiagonals similarly,
C and finally its highest superdiagonal in the first N+1-MB
C positions of the last column. Contents of storage locations
C not part of the matrix are arbitrary. A is a two-dimensional
C REAL array, dimensioned A(NM,MBW).
C
C E21 specifies the ordering of the eigenvalues and contains
C 0.0E0 if the eigenvalues are in ascending order, or
C 2.0E0 if the eigenvalues are in descending order.
C If the subroutine is being used to solve systems of linear
C equations, E21 should be set to 1.0E0 if the coefficient
C matrix is symmetric and to -1.0E0 if not. E21 is a REAL
C variable.
C
C M is the number of specified eigenvalues or the number of
C systems of linear equations. M is an INTEGER variable.
C
C W contains the M eigenvalues in ascending or descending order.
C If the subroutine is being used to solve systems of linear
C equations (A-W(J)*I)*X(J)=B(J), where I is the identity
C matrix, W(J) should be set accordingly, for J=1,2,...,M.
C W is a one-dimensional REAL array, dimensioned W(M).
C
C Z contains the constant matrix columns (B(J),J=1,2,...,M), if
C the subroutine is used to solve systems of linear equations.
C Z is a two-dimensional REAL array, dimensioned Z(NM,M).
C
C NV must be set to the dimension of the array parameter RV
C as declared in the calling program dimension statement.
C NV is an INTEGER variable.
C
C On OUTPUT
C
C A and W are unaltered.
C
C Z contains the associated set of orthogonal eigenvectors.
C Any vector which fails to converge is set to zero. If the
C subroutine is used to solve systems of linear equations,
C Z contains the solution matrix columns (X(J),J=1,2,...,M).
C
C IERR is an INTEGER flag set to
C Zero for normal return,
C -J if the eigenvector corresponding to the J-th
C eigenvalue fails to converge, or if the J-th
C system of linear equations is nearly singular.
C
C RV and RV6 are temporary storage arrays. If the subroutine
C is being used to solve systems of linear equations, the
C determinant (up to sign) of A-W(M)*I is available, upon
C return, as the product of the first N elements of RV.
C RV and RV6 are one-dimensional REAL arrays. Note that RV
C is dimensioned RV(NV), where NV must be at least N*(2*MB-1).
C RV6 is dimensioned RV6(N).
C
C Questions and comments should be directed to B. S. Garbow,
C Applied Mathematics Division, ARGONNE NATIONAL LABORATORY
C ------------------------------------------------------------------
C
C***REFERENCES B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow,
C Y. Ikebe, V. C. Klema and C. B. Moler, Matrix Eigen-
C system Routines - EISPACK Guide, Springer-Verlag,
C 1976.
C***ROUTINES CALLED (NONE)
C***REVISION HISTORY (YYMMDD)
C 760101 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
C 890831 Modified array declarations. (WRB)
C 890831 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE BANDV
C
INTEGER I,J,K,M,N,R,II,IJ,JJ,KJ,MB,M1,NM,NV,IJ1,ITS,KJ1,MBW,M21
INTEGER IERR,MAXJ,MAXK,GROUP
REAL A(NM,*),W(*),Z(NM,*),RV(*),RV6(*)
REAL U,V,UK,XU,X0,X1,E21,EPS2,EPS3,EPS4,NORM,ORDER,S
C
C***FIRST EXECUTABLE STATEMENT BANDV
IERR = 0
IF (M .EQ. 0) GO TO 1001
MB = MBW
IF (E21 .LT. 0.0E0) MB = (MBW + 1) / 2
M1 = MB - 1
M21 = M1 + MB
ORDER = 1.0E0 - ABS(E21)
C .......... FIND VECTORS BY INVERSE ITERATION ..........
DO 920 R = 1, M
ITS = 1
X1 = W(R)
IF (R .NE. 1) GO TO 100
C .......... COMPUTE NORM OF MATRIX ..........
NORM = 0.0E0
C
DO 60 J = 1, MB
JJ = MB + 1 - J
KJ = JJ + M1
IJ = 1
S = 0.0E0
C
DO 40 I = JJ, N
S = S + ABS(A(I,J))
IF (E21 .GE. 0.0E0) GO TO 40
S = S + ABS(A(IJ,KJ))
IJ = IJ + 1
40 CONTINUE
C
NORM = MAX(NORM,S)
60 CONTINUE
C
IF (E21 .LT. 0.0E0) NORM = 0.5E0 * NORM
C .......... EPS2 IS THE CRITERION FOR GROUPING,
C EPS3 REPLACES ZERO PIVOTS AND EQUAL
C ROOTS ARE MODIFIED BY EPS3,
C EPS4 IS TAKEN VERY SMALL TO AVOID OVERFLOW ..........
IF (NORM .EQ. 0.0E0) NORM = 1.0E0
EPS2 = 1.0E-3 * NORM * ABS(ORDER)
EPS3 = NORM
70 EPS3 = 0.5E0*EPS3
IF (NORM + EPS3 .GT. NORM) GO TO 70
UK = SQRT(REAL(N))
EPS3 = UK * EPS3
EPS4 = UK * EPS3
80 GROUP = 0
GO TO 120
C .......... LOOK FOR CLOSE OR COINCIDENT ROOTS ..........
100 IF (ABS(X1-X0) .GE. EPS2) GO TO 80
GROUP = GROUP + 1
IF (ORDER * (X1 - X0) .LE. 0.0E0) X1 = X0 + ORDER * EPS3
C .......... EXPAND MATRIX, SUBTRACT EIGENVALUE,
C AND INITIALIZE VECTOR ..........
120 DO 200 I = 1, N
IJ = I + MIN(0,I-M1) * N
KJ = IJ + MB * N
IJ1 = KJ + M1 * N
IF (M1 .EQ. 0) GO TO 180
C
DO 150 J = 1, M1
IF (IJ .GT. M1) GO TO 125
IF (IJ .GT. 0) GO TO 130
RV(IJ1) = 0.0E0
IJ1 = IJ1 + N
GO TO 130
125 RV(IJ) = A(I,J)
130 IJ = IJ + N
II = I + J
IF (II .GT. N) GO TO 150
JJ = MB - J
IF (E21 .GE. 0.0E0) GO TO 140
II = I
JJ = MB + J
140 RV(KJ) = A(II,JJ)
KJ = KJ + N
150 CONTINUE
C
180 RV(IJ) = A(I,MB) - X1
RV6(I) = EPS4
IF (ORDER .EQ. 0.0E0) RV6(I) = Z(I,R)
200 CONTINUE
C
IF (M1 .EQ. 0) GO TO 600
C .......... ELIMINATION WITH INTERCHANGES ..........
DO 580 I = 1, N
II = I + 1
MAXK = MIN(I+M1-1,N)
MAXJ = MIN(N-I,M21-2) * N
C
DO 360 K = I, MAXK
KJ1 = K
J = KJ1 + N
JJ = J + MAXJ
C
DO 340 KJ = J, JJ, N
RV(KJ1) = RV(KJ)
KJ1 = KJ
340 CONTINUE
C
RV(KJ1) = 0.0E0
360 CONTINUE
C
IF (I .EQ. N) GO TO 580
U = 0.0E0
MAXK = MIN(I+M1,N)
MAXJ = MIN(N-II,M21-2) * N
C
DO 450 J = I, MAXK
IF (ABS(RV(J)) .LT. ABS(U)) GO TO 450
U = RV(J)
K = J
450 CONTINUE
C
J = I + N
JJ = J + MAXJ
IF (K .EQ. I) GO TO 520
KJ = K
C
DO 500 IJ = I, JJ, N
V = RV(IJ)
RV(IJ) = RV(KJ)
RV(KJ) = V
KJ = KJ + N
500 CONTINUE
C
IF (ORDER .NE. 0.0E0) GO TO 520
V = RV6(I)
RV6(I) = RV6(K)
RV6(K) = V
520 IF (U .EQ. 0.0E0) GO TO 580
C
DO 560 K = II, MAXK
V = RV(K) / U
KJ = K
C
DO 540 IJ = J, JJ, N
KJ = KJ + N
RV(KJ) = RV(KJ) - V * RV(IJ)
540 CONTINUE
C
IF (ORDER .EQ. 0.0E0) RV6(K) = RV6(K) - V * RV6(I)
560 CONTINUE
C
580 CONTINUE
C .......... BACK SUBSTITUTION
C FOR I=N STEP -1 UNTIL 1 DO -- ..........
600 DO 630 II = 1, N
I = N + 1 - II
MAXJ = MIN(II,M21)
IF (MAXJ .EQ. 1) GO TO 620
IJ1 = I
J = IJ1 + N
JJ = J + (MAXJ - 2) * N
C
DO 610 IJ = J, JJ, N
IJ1 = IJ1 + 1
RV6(I) = RV6(I) - RV(IJ) * RV6(IJ1)
610 CONTINUE
C
620 V = RV(I)
IF (ABS(V) .GE. EPS3) GO TO 625
C .......... SET ERROR -- NEARLY SINGULAR LINEAR SYSTEM ..........
IF (ORDER .EQ. 0.0E0) IERR = -R
V = SIGN(EPS3,V)
625 RV6(I) = RV6(I) / V
630 CONTINUE
C
XU = 1.0E0
IF (ORDER .EQ. 0.0E0) GO TO 870
C .......... ORTHOGONALIZE WITH RESPECT TO PREVIOUS
C MEMBERS OF GROUP ..........
IF (GROUP .EQ. 0) GO TO 700
C
DO 680 JJ = 1, GROUP
J = R - GROUP - 1 + JJ
XU = 0.0E0
C
DO 640 I = 1, N
640 XU = XU + RV6(I) * Z(I,J)
C
DO 660 I = 1, N
660 RV6(I) = RV6(I) - XU * Z(I,J)
C
680 CONTINUE
C
700 NORM = 0.0E0
C
DO 720 I = 1, N
720 NORM = NORM + ABS(RV6(I))
C
IF (NORM .GE. 0.1E0) GO TO 840
C .......... IN-LINE PROCEDURE FOR CHOOSING
C A NEW STARTING VECTOR ..........
IF (ITS .GE. N) GO TO 830
ITS = ITS + 1
XU = EPS4 / (UK + 1.0E0)
RV6(1) = EPS4
C
DO 760 I = 2, N
760 RV6(I) = XU
C
RV6(ITS) = RV6(ITS) - EPS4 * UK
GO TO 600
C .......... SET ERROR -- NON-CONVERGED EIGENVECTOR ..........
830 IERR = -R
XU = 0.0E0
GO TO 870
C .......... NORMALIZE SO THAT SUM OF SQUARES IS
C 1 AND EXPAND TO FULL ORDER ..........
840 U = 0.0E0
C
DO 860 I = 1, N
860 U = U + RV6(I)**2
C
XU = 1.0E0 / SQRT(U)
C
870 DO 900 I = 1, N
900 Z(I,R) = RV6(I) * XU
C
X0 = X1
920 CONTINUE
C
1001 RETURN
END