+++ /dev/null
-*DECK TQL2
- SUBROUTINE TQL2 (NM, N, D, E, Z, IERR)
-C***BEGIN PROLOGUE TQL2
-C***PURPOSE Compute the eigenvalues and eigenvectors of symmetric
-C tridiagonal matrix.
-C***LIBRARY SLATEC (EISPACK)
-C***CATEGORY D4A5, D4C2A
-C***TYPE SINGLE PRECISION (TQL2-S)
-C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK
-C***AUTHOR Smith, B. T., et al.
-C***DESCRIPTION
-C
-C This subroutine is a translation of the ALGOL procedure TQL2,
-C NUM. MATH. 11, 293-306(1968) by Bowdler, Martin, Reinsch, and
-C Wilkinson.
-C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 227-240(1971).
-C
-C This subroutine finds the eigenvalues and eigenvectors
-C of a SYMMETRIC TRIDIAGONAL matrix by the QL method.
-C The eigenvectors of a FULL SYMMETRIC matrix can also
-C be found if TRED2 has been used to reduce this
-C full matrix to tridiagonal form.
-C
-C On Input
-C
-C NM must be set to the row dimension of the two-dimensional
-C array parameter, Z, as declared in the calling program
-C dimension statement. NM is an INTEGER variable.
-C
-C N is the order of the matrix. N is an INTEGER variable.
-C N must be less than or equal to NM.
-C
-C D contains the diagonal elements of the symmetric tridiagonal
-C matrix. D is a one-dimensional REAL array, dimensioned D(N).
-C
-C E contains the subdiagonal elements of the symmetric
-C tridiagonal matrix in its last N-1 positions. E(1) is
-C arbitrary. E is a one-dimensional REAL array, dimensioned
-C E(N).
-C
-C Z contains the transformation matrix produced in the
-C reduction by TRED2, if performed. If the eigenvectors
-C of the tridiagonal matrix are desired, Z must contain
-C the identity matrix. Z is a two-dimensional REAL array,
-C dimensioned Z(NM,N).
-C
-C On Output
-C
-C D contains the eigenvalues in ascending order. If an
-C error exit is made, the eigenvalues are correct but
-C unordered for indices 1, 2, ..., IERR-1.
-C
-C E has been destroyed.
-C
-C Z contains orthonormal eigenvectors of the symmetric
-C tridiagonal (or full) matrix. If an error exit is made,
-C Z contains the eigenvectors associated with the stored
-C eigenvalues.
-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 30 iterations.
-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 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 TQL2
-C
- INTEGER I,J,K,L,M,N,II,L1,L2,NM,MML,IERR
- REAL D(*),E(*),Z(NM,*)
- REAL B,C,C2,C3,DL1,EL1,F,G,H,P,R,S,S2
- REAL PYTHAG
-C
-C***FIRST EXECUTABLE STATEMENT TQL2
- IERR = 0
- IF (N .EQ. 1) GO TO 1001
-C
- DO 100 I = 2, N
- 100 E(I-1) = E(I)
-C
- F = 0.0E0
- B = 0.0E0
- E(N) = 0.0E0
-C
- DO 240 L = 1, N
- J = 0
- H = ABS(D(L)) + ABS(E(L))
- IF (B .LT. H) B = H
-C .......... LOOK FOR SMALL SUB-DIAGONAL ELEMENT ..........
- DO 110 M = L, N
- IF (B + ABS(E(M)) .EQ. B) GO TO 120
-C .......... E(N) IS ALWAYS ZERO, SO THERE IS NO EXIT
-C THROUGH THE BOTTOM OF THE LOOP ..........
- 110 CONTINUE
-C
- 120 IF (M .EQ. L) GO TO 220
- 130 IF (J .EQ. 30) GO TO 1000
- J = J + 1
-C .......... FORM SHIFT ..........
- L1 = L + 1
- L2 = L1 + 1
- G = D(L)
- P = (D(L1) - G) / (2.0E0 * E(L))
- R = PYTHAG(P,1.0E0)
- D(L) = E(L) / (P + SIGN(R,P))
- D(L1) = E(L) * (P + SIGN(R,P))
- DL1 = D(L1)
- H = G - D(L)
- IF (L2 .GT. N) GO TO 145
-C
- DO 140 I = L2, N
- 140 D(I) = D(I) - H
-C
- 145 F = F + H
-C .......... QL TRANSFORMATION ..........
- P = D(M)
- C = 1.0E0
- C2 = C
- EL1 = E(L1)
- S = 0.0E0
- MML = M - L
-C .......... FOR I=M-1 STEP -1 UNTIL L DO -- ..........
- DO 200 II = 1, MML
- C3 = C2
- C2 = C
- S2 = S
- I = M - II
- G = C * E(I)
- H = C * P
- IF (ABS(P) .LT. ABS(E(I))) GO TO 150
- C = E(I) / P
- R = SQRT(C*C+1.0E0)
- E(I+1) = S * P * R
- S = C / R
- C = 1.0E0 / R
- GO TO 160
- 150 C = P / E(I)
- R = SQRT(C*C+1.0E0)
- E(I+1) = S * E(I) * R
- S = 1.0E0 / R
- C = C * S
- 160 P = C * D(I) - S * G
- D(I+1) = H + S * (C * G + S * D(I))
-C .......... FORM VECTOR ..........
- DO 180 K = 1, N
- H = Z(K,I+1)
- Z(K,I+1) = S * Z(K,I) + C * H
- Z(K,I) = C * Z(K,I) - S * H
- 180 CONTINUE
-C
- 200 CONTINUE
-C
- P = -S * S2 * C3 * EL1 * E(L) / DL1
- E(L) = S * P
- D(L) = C * P
- IF (B + ABS(E(L)) .GT. B) GO TO 130
- 220 D(L) = D(L) + F
- 240 CONTINUE
-C .......... ORDER EIGENVALUES AND EIGENVECTORS ..........
- DO 300 II = 2, N
- I = II - 1
- K = I
- P = D(I)
-C
- DO 260 J = II, N
- IF (D(J) .GE. P) GO TO 260
- K = J
- P = D(J)
- 260 CONTINUE
-C
- IF (K .EQ. I) GO TO 300
- D(K) = D(I)
- D(I) = P
-C
- DO 280 J = 1, N
- P = Z(J,I)
- Z(J,I) = Z(J,K)
- Z(J,K) = P
- 280 CONTINUE
-C
- 300 CONTINUE
-C
- GO TO 1001
-C .......... SET ERROR -- NO CONVERGENCE TO AN
-C EIGENVALUE AFTER 30 ITERATIONS ..........
- 1000 IERR = L
- 1001 RETURN
- END
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