*DECK BQR
SUBROUTINE BQR (NM, N, MB, A, T, R, IERR, NV, RV)
C***BEGIN PROLOGUE BQR
C***PURPOSE Compute some of the eigenvalues of a real symmetric
C matrix using the QR method with shifts of origin.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4A6
C***TYPE SINGLE PRECISION (BQR-S)
C***KEYWORDS EIGENVALUES, EISPACK
C***AUTHOR Smith, B. T., et al.
C***DESCRIPTION
C
C This subroutine is a translation of the ALGOL procedure BQR,
C NUM. MATH. 16, 85-92(1970) by Martin, Reinsch, and Wilkinson.
C HANDBOOK FOR AUTO. COMP., VOL II-LINEAR ALGEBRA, 266-272(1971).
C
C This subroutine finds the eigenvalue of smallest (usually)
C magnitude of a REAL SYMMETRIC BAND matrix using the
C QR algorithm with shifts of origin. Consecutive calls
C can be made to find further eigenvalues.
C
C On INPUT
C
C NM must be set to the row dimension of the two-dimensional
C array parameter, A, as declared in the calling program
C 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 MB is the (half) band width of the matrix, defined as the
C number of adjacent diagonals, including the principal
C diagonal, required to specify the non-zero portion of the
C lower triangle of the matrix. MB is an INTEGER variable.
C MB must be less than or equal to N on first call.
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 the last column.
C Contents of storages not part of the matrix are arbitrary.
C On a subsequent call, its output contents from the previous
C call should be passed. A is a two-dimensional REAL array,
C dimensioned A(NM,MB).
C
C T specifies the shift (of eigenvalues) applied to the diagonal
C of A in forming the input matrix. What is actually determined
C is the eigenvalue of A+TI (I is the identity matrix) nearest
C to T. On a subsequent call, the output value of T from the
C previous call should be passed if the next nearest eigenvalue
C is sought. T is a REAL variable.
C
C R should be specified as zero on the first call, and as its
C output value from the previous call on a subsequent call.
C It is used to determine when the last row and column of
C the transformed band matrix can be regarded as negligible.
C R is a REAL variable.
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 contains the transformed band matrix. The matrix A+TI
C derived from the output parameters is similar to the
C input A+TI to within rounding errors. Its last row and
C column are null (if IERR is zero).
C
C T contains the computed eigenvalue of A+TI (if IERR is zero),
C where I is the identity matrix.
C
C R contains the maximum of its input value and the norm of the
C last column of the input matrix A.
C
C IERR is an INTEGER flag set to
C Zero for normal return,
C J if the J-th eigenvalue has not been
C determined after a total of 30 iterations.
C
C RV is a one-dimensional REAL array of dimension NV which is
C at least (2*MB**2+4*MB-3), used for temporary storage. The
C first (3*MB-2) locations correspond to the ALGOL array B,
C the next (2*MB-1) locations correspond to the ALGOL array H,
C and the final (2*MB**2-MB) locations correspond to the MB
C by (2*MB-1) ALGOL array U.
C
C NOTE. For a subsequent call, N should be replaced by N-1, but
C MB should not be altered even when it exceeds the current N.
C
C Calls PYTHAG(A,B) for SQRT(A**2 + B**2).
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 PYTHAG
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 BQR
C
INTEGER I,J,K,L,M,N,II,IK,JK,JM,KJ,KK,KM,LL,MB,MK,MN,MZ
INTEGER M1,M2,M3,M4,NI,NM,NV,ITS,KJ1,M21,M31,IERR,IMULT
REAL A(NM,*),RV(*)
REAL F,G,Q,R,S,T,SCALE
REAL PYTHAG
C
C***FIRST EXECUTABLE STATEMENT BQR
IERR = 0
M1 = MIN(MB,N)
M = M1 - 1
M2 = M + M
M21 = M2 + 1
M3 = M21 + M
M31 = M3 + 1
M4 = M31 + M2
MN = M + N
MZ = MB - M1
ITS = 0
C .......... TEST FOR CONVERGENCE ..........
40 G = A(N,MB)
IF (M .EQ. 0) GO TO 360
F = 0.0E0
C
DO 50 K = 1, M
MK = K + MZ
F = F + ABS(A(N,MK))
50 CONTINUE
C
IF (ITS .EQ. 0 .AND. F .GT. R) R = F
IF (R + F .LE. R) GO TO 360
IF (ITS .EQ. 30) GO TO 1000
ITS = ITS + 1
C .......... FORM SHIFT FROM BOTTOM 2 BY 2 MINOR ..........
IF (F .GT. 0.25E0 * R .AND. ITS .LT. 5) GO TO 90
F = A(N,MB-1)
IF (F .EQ. 0.0E0) GO TO 70
Q = (A(N-1,MB) - G) / (2.0E0 * F)
S = PYTHAG(Q,1.0E0)
G = G - F / (Q + SIGN(S,Q))
70 T = T + G
C
DO 80 I = 1, N
80 A(I,MB) = A(I,MB) - G
C
90 DO 100 K = M31, M4
100 RV(K) = 0.0E0
C
DO 350 II = 1, MN
I = II - M
NI = N - II
IF (NI .LT. 0) GO TO 230
C .......... FORM COLUMN OF SHIFTED MATRIX A-G*I ..........
L = MAX(1,2-I)
C
DO 110 K = 1, M3
110 RV(K) = 0.0E0
C
DO 120 K = L, M1
KM = K + M
MK = K + MZ
RV(KM) = A(II,MK)
120 CONTINUE
C
LL = MIN(M,NI)
IF (LL .EQ. 0) GO TO 135
C
DO 130 K = 1, LL
KM = K + M21
IK = II + K
MK = MB - K
RV(KM) = A(IK,MK)
130 CONTINUE
C .......... PRE-MULTIPLY WITH HOUSEHOLDER REFLECTIONS ..........
135 LL = M2
IMULT = 0
C .......... MULTIPLICATION PROCEDURE ..........
140 KJ = M4 - M1
C
DO 170 J = 1, LL
KJ = KJ + M1
JM = J + M3
IF (RV(JM) .EQ. 0.0E0) GO TO 170
F = 0.0E0
C
DO 150 K = 1, M1
KJ = KJ + 1
JK = J + K - 1
F = F + RV(KJ) * RV(JK)
150 CONTINUE
C
F = F / RV(JM)
KJ = KJ - M1
C
DO 160 K = 1, M1
KJ = KJ + 1
JK = J + K - 1
RV(JK) = RV(JK) - RV(KJ) * F
160 CONTINUE
C
KJ = KJ - M1
170 CONTINUE
C
IF (IMULT .NE. 0) GO TO 280
C .......... HOUSEHOLDER REFLECTION ..........
F = RV(M21)
S = 0.0E0
RV(M4) = 0.0E0
SCALE = 0.0E0
C
DO 180 K = M21, M3
180 SCALE = SCALE + ABS(RV(K))
C
IF (SCALE .EQ. 0.0E0) GO TO 210
C
DO 190 K = M21, M3
190 S = S + (RV(K)/SCALE)**2
C
S = SCALE * SCALE * S
G = -SIGN(SQRT(S),F)
RV(M21) = G
RV(M4) = S - F * G
KJ = M4 + M2 * M1 + 1
RV(KJ) = F - G
C
DO 200 K = 2, M1
KJ = KJ + 1
KM = K + M2
RV(KJ) = RV(KM)
200 CONTINUE
C .......... SAVE COLUMN OF TRIANGULAR FACTOR R ..........
210 DO 220 K = L, M1
KM = K + M
MK = K + MZ
A(II,MK) = RV(KM)
220 CONTINUE
C
230 L = MAX(1,M1+1-I)
IF (I .LE. 0) GO TO 300
C .......... PERFORM ADDITIONAL STEPS ..........
DO 240 K = 1, M21
240 RV(K) = 0.0E0
C
LL = MIN(M1,NI+M1)
C .......... GET ROW OF TRIANGULAR FACTOR R ..........
DO 250 KK = 1, LL
K = KK - 1
KM = K + M1
IK = I + K
MK = MB - K
RV(KM) = A(IK,MK)
250 CONTINUE
C .......... POST-MULTIPLY WITH HOUSEHOLDER REFLECTIONS ..........
LL = M1
IMULT = 1
GO TO 140
C .......... STORE COLUMN OF NEW A MATRIX ..........
280 DO 290 K = L, M1
MK = K + MZ
A(I,MK) = RV(K)
290 CONTINUE
C .......... UPDATE HOUSEHOLDER REFLECTIONS ..........
300 IF (L .GT. 1) L = L - 1
KJ1 = M4 + L * M1
C
DO 320 J = L, M2
JM = J + M3
RV(JM) = RV(JM+1)
C
DO 320 K = 1, M1
KJ1 = KJ1 + 1
KJ = KJ1 - M1
RV(KJ) = RV(KJ1)
320 CONTINUE
C
350 CONTINUE
C
GO TO 40
C .......... CONVERGENCE ..........
360 T = T + G
C
DO 380 I = 1, N
380 A(I,MB) = A(I,MB) - G
C
DO 400 K = 1, M1
MK = K + MZ
A(N,MK) = 0.0E0
400 CONTINUE
C
GO TO 1001
C .......... SET ERROR -- NO CONVERGENCE TO
C EIGENVALUE AFTER 30 ITERATIONS ..........
1000 IERR = N
1001 RETURN
END