*DECK CORTH SUBROUTINE CORTH (NM, N, LOW, IGH, AR, AI, ORTR, ORTI) C***BEGIN PROLOGUE CORTH C***PURPOSE Reduce a complex general matrix to complex upper Hessenberg C form using unitary similarity transformations. C***LIBRARY SLATEC (EISPACK) C***CATEGORY D4C1B2 C***TYPE COMPLEX (ORTHES-S, CORTH-C) 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 ORTHES, NUM. MATH. 12, 349-368(1968) C by Martin and Wilkinson. C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 339-358(1971). C C Given a COMPLEX GENERAL matrix, this subroutine C reduces a submatrix situated in rows and columns C LOW through IGH to upper Hessenberg form by C unitary similarity transformations. C C On INPUT C C NM must be set to the row dimension of the two-dimensional C array parameters, AR and AI, as declared in the calling C program dimension statement. NM is an INTEGER variable. C C N is the order of the matrix A=(AR,AI). N is an INTEGER C variable. N must be less than or equal to NM. C C LOW and IGH are two INTEGER variables determined by the C balancing subroutine CBAL. If CBAL has not been used, C set LOW=1 and IGH equal to the order of the matrix, N. C C AR and AI contain the real and imaginary parts, respectively, C of the complex input matrix. AR and AI are two-dimensional C REAL arrays, dimensioned AR(NM,N) and AI(NM,N). C C On OUTPUT C C AR and AI contain the real and imaginary parts, respectively, C of the Hessenberg matrix. Information about the unitary C transformations used in the reduction is stored in the C remaining triangles under the Hessenberg matrix. C C ORTR and ORTI contain further information about the unitary C transformations. Only elements LOW through IGH are used. C ORTR and ORTI are one-dimensional REAL arrays, dimensioned C ORTR(IGH) and ORTI(IGH). 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 CORTH C INTEGER I,J,M,N,II,JJ,LA,MP,NM,IGH,KP1,LOW REAL AR(NM,*),AI(NM,*),ORTR(*),ORTI(*) REAL F,G,H,FI,FR,SCALE REAL PYTHAG C C***FIRST EXECUTABLE STATEMENT CORTH LA = IGH - 1 KP1 = LOW + 1 IF (LA .LT. KP1) GO TO 200 C DO 180 M = KP1, LA H = 0.0E0 ORTR(M) = 0.0E0 ORTI(M) = 0.0E0 SCALE = 0.0E0 C .......... SCALE COLUMN (ALGOL TOL THEN NOT NEEDED) .......... DO 90 I = M, IGH 90 SCALE = SCALE + ABS(AR(I,M-1)) + ABS(AI(I,M-1)) C IF (SCALE .EQ. 0.0E0) GO TO 180 MP = M + IGH C .......... FOR I=IGH STEP -1 UNTIL M DO -- .......... DO 100 II = M, IGH I = MP - II ORTR(I) = AR(I,M-1) / SCALE ORTI(I) = AI(I,M-1) / SCALE H = H + ORTR(I) * ORTR(I) + ORTI(I) * ORTI(I) 100 CONTINUE C G = SQRT(H) F = PYTHAG(ORTR(M),ORTI(M)) IF (F .EQ. 0.0E0) GO TO 103 H = H + F * G G = G / F ORTR(M) = (1.0E0 + G) * ORTR(M) ORTI(M) = (1.0E0 + G) * ORTI(M) GO TO 105 C 103 ORTR(M) = G AR(M,M-1) = SCALE C .......... FORM (I-(U*UT)/H) * A .......... 105 DO 130 J = M, N FR = 0.0E0 FI = 0.0E0 C .......... FOR I=IGH STEP -1 UNTIL M DO -- .......... DO 110 II = M, IGH I = MP - II FR = FR + ORTR(I) * AR(I,J) + ORTI(I) * AI(I,J) FI = FI + ORTR(I) * AI(I,J) - ORTI(I) * AR(I,J) 110 CONTINUE C FR = FR / H FI = FI / H C DO 120 I = M, IGH AR(I,J) = AR(I,J) - FR * ORTR(I) + FI * ORTI(I) AI(I,J) = AI(I,J) - FR * ORTI(I) - FI * ORTR(I) 120 CONTINUE C 130 CONTINUE C .......... FORM (I-(U*UT)/H)*A*(I-(U*UT)/H) .......... DO 160 I = 1, IGH FR = 0.0E0 FI = 0.0E0 C .......... FOR J=IGH STEP -1 UNTIL M DO -- .......... DO 140 JJ = M, IGH J = MP - JJ FR = FR + ORTR(J) * AR(I,J) - ORTI(J) * AI(I,J) FI = FI + ORTR(J) * AI(I,J) + ORTI(J) * AR(I,J) 140 CONTINUE C FR = FR / H FI = FI / H C DO 150 J = M, IGH AR(I,J) = AR(I,J) - FR * ORTR(J) - FI * ORTI(J) AI(I,J) = AI(I,J) + FR * ORTI(J) - FI * ORTR(J) 150 CONTINUE C 160 CONTINUE C ORTR(M) = SCALE * ORTR(M) ORTI(M) = SCALE * ORTI(M) AR(M,M-1) = -G * AR(M,M-1) AI(M,M-1) = -G * AI(M,M-1) 180 CONTINUE C 200 RETURN END