*DECK HTRID3 SUBROUTINE HTRID3 (NM, N, A, D, E, E2, TAU) C***BEGIN PROLOGUE HTRID3 C***PURPOSE Reduce a complex Hermitian (packed) matrix to a real C symmetric tridiagonal matrix by unitary similarity C transformations. C***LIBRARY SLATEC (EISPACK) C***CATEGORY D4C1B1 C***TYPE SINGLE PRECISION (HTRID3-S) C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK C***AUTHOR Smith, B. T., et al. C***DESCRIPTION C C This subroutine is a translation of a complex analogue of C the ALGOL procedure TRED3, NUM. MATH. 11, 181-195(1968) C by Martin, Reinsch, and Wilkinson. C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971). C C This subroutine reduces a COMPLEX HERMITIAN matrix, stored as C a single square array, to a real symmetric tridiagonal matrix C using unitary similarity transformations. 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. N is an INTEGER variable. C N must be less than or equal to NM. C C A contains the lower triangle of the complex Hermitian input C matrix. The real parts of the matrix elements are stored C in the full lower triangle of A, and the imaginary parts C are stored in the transposed positions of the strict upper C triangle of A. No storage is required for the zero C imaginary parts of the diagonal elements. A is a two- C dimensional REAL array, dimensioned A(NM,N). C C On OUTPUT C C A contains some information about the unitary transformations C used in the reduction. C C D contains the diagonal elements of the real symmetric C tridiagonal matrix. D is a one-dimensional REAL array, C dimensioned D(N). C C E contains the subdiagonal elements of the real tridiagonal C matrix in its last N-1 positions. E(1) is set to zero. C E is a one-dimensional REAL array, dimensioned E(N). C C E2 contains the squares of the corresponding elements of E. C E2(1) is set to zero. E2 may coincide with E if the squares C are not needed. E2 is a one-dimensional REAL array, C dimensioned E2(N). C C TAU contains further information about the transformations. C TAU is a one-dimensional REAL array, dimensioned TAU(2,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 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 HTRID3 C INTEGER I,J,K,L,N,II,NM,JM1,JP1 REAL A(NM,*),D(*),E(*),E2(*),TAU(2,*) REAL F,G,H,FI,GI,HH,SI,SCALE REAL PYTHAG C C***FIRST EXECUTABLE STATEMENT HTRID3 TAU(1,N) = 1.0E0 TAU(2,N) = 0.0E0 C .......... FOR I=N STEP -1 UNTIL 1 DO -- .......... DO 300 II = 1, N I = N + 1 - II L = I - 1 H = 0.0E0 SCALE = 0.0E0 IF (L .LT. 1) GO TO 130 C .......... SCALE ROW (ALGOL TOL THEN NOT NEEDED) .......... DO 120 K = 1, L 120 SCALE = SCALE + ABS(A(I,K)) + ABS(A(K,I)) C IF (SCALE .NE. 0.0E0) GO TO 140 TAU(1,L) = 1.0E0 TAU(2,L) = 0.0E0 130 E(I) = 0.0E0 E2(I) = 0.0E0 GO TO 290 C 140 DO 150 K = 1, L A(I,K) = A(I,K) / SCALE A(K,I) = A(K,I) / SCALE H = H + A(I,K) * A(I,K) + A(K,I) * A(K,I) 150 CONTINUE C E2(I) = SCALE * SCALE * H G = SQRT(H) E(I) = SCALE * G F = PYTHAG(A(I,L),A(L,I)) C .......... FORM NEXT DIAGONAL ELEMENT OF MATRIX T .......... IF (F .EQ. 0.0E0) GO TO 160 TAU(1,L) = (A(L,I) * TAU(2,I) - A(I,L) * TAU(1,I)) / F SI = (A(I,L) * TAU(2,I) + A(L,I) * TAU(1,I)) / F H = H + F * G G = 1.0E0 + G / F A(I,L) = G * A(I,L) A(L,I) = G * A(L,I) IF (L .EQ. 1) GO TO 270 GO TO 170 160 TAU(1,L) = -TAU(1,I) SI = TAU(2,I) A(I,L) = G 170 F = 0.0E0 C DO 240 J = 1, L G = 0.0E0 GI = 0.0E0 IF (J .EQ. 1) GO TO 190 JM1 = J - 1 C .......... FORM ELEMENT OF A*U .......... DO 180 K = 1, JM1 G = G + A(J,K) * A(I,K) + A(K,J) * A(K,I) GI = GI - A(J,K) * A(K,I) + A(K,J) * A(I,K) 180 CONTINUE C 190 G = G + A(J,J) * A(I,J) GI = GI - A(J,J) * A(J,I) JP1 = J + 1 IF (L .LT. JP1) GO TO 220 C DO 200 K = JP1, L G = G + A(K,J) * A(I,K) - A(J,K) * A(K,I) GI = GI - A(K,J) * A(K,I) - A(J,K) * A(I,K) 200 CONTINUE C .......... FORM ELEMENT OF P .......... 220 E(J) = G / H TAU(2,J) = GI / H F = F + E(J) * A(I,J) - TAU(2,J) * A(J,I) 240 CONTINUE C HH = F / (H + H) C .......... FORM REDUCED A .......... DO 260 J = 1, L F = A(I,J) G = E(J) - HH * F E(J) = G FI = -A(J,I) GI = TAU(2,J) - HH * FI TAU(2,J) = -GI A(J,J) = A(J,J) - 2.0E0 * (F * G + FI * GI) IF (J .EQ. 1) GO TO 260 JM1 = J - 1 C DO 250 K = 1, JM1 A(J,K) = A(J,K) - F * E(K) - G * A(I,K) 1 + FI * TAU(2,K) + GI * A(K,I) A(K,J) = A(K,J) - F * TAU(2,K) - G * A(K,I) 1 - FI * E(K) - GI * A(I,K) 250 CONTINUE C 260 CONTINUE C 270 DO 280 K = 1, L A(I,K) = SCALE * A(I,K) A(K,I) = SCALE * A(K,I) 280 CONTINUE C TAU(2,L) = -SI 290 D(I) = A(I,I) A(I,I) = SCALE * SQRT(H) 300 CONTINUE C RETURN END