*DECK ORTHES SUBROUTINE ORTHES (NM, N, LOW, IGH, A, ORT) C***BEGIN PROLOGUE ORTHES C***PURPOSE Reduce a real general matrix to upper Hessenberg form C using orthogonal similarity transformations. C***LIBRARY SLATEC (EISPACK) C***CATEGORY D4C1B2 C***TYPE SINGLE PRECISION (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 the ALGOL procedure ORTHES, C NUM. MATH. 12, 349-368(1968) by Martin and Wilkinson. C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 339-358(1971). C C Given a REAL GENERAL matrix, this subroutine C reduces a submatrix situated in rows and columns C LOW through IGH to upper Hessenberg form by C orthogonal 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 A. N is an INTEGER variable. C N must be less than or equal to NM. C C LOW and IGH are two INTEGER variables determined by the C balancing subroutine BALANC. If BALANC has not been C used, set LOW=1 and IGH equal to the order of the matrix, N. C C A contains the general matrix to be reduced to upper C Hessenberg form. A is a two-dimensional REAL array, C dimensioned A(NM,N). C C On OUTPUT C C A contains the upper Hessenberg matrix. Some information about C the orthogonal transformations used in the reduction C is stored in the remaining triangle under the Hessenberg C matrix. C C ORT contains further information about the orthogonal trans- C formations used in the reduction. Only elements LOW+1 C through IGH are used. ORT is a one-dimensional REAL array, C dimensioned ORT(IGH). 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 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 ORTHES C INTEGER I,J,M,N,II,JJ,LA,MP,NM,IGH,KP1,LOW REAL A(NM,*),ORT(*) REAL F,G,H,SCALE C C***FIRST EXECUTABLE STATEMENT ORTHES LA = IGH - 1 KP1 = LOW + 1 IF (LA .LT. KP1) GO TO 200 C DO 180 M = KP1, LA H = 0.0E0 ORT(M) = 0.0E0 SCALE = 0.0E0 C .......... SCALE COLUMN (ALGOL TOL THEN NOT NEEDED) .......... DO 90 I = M, IGH 90 SCALE = SCALE + ABS(A(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 ORT(I) = A(I,M-1) / SCALE H = H + ORT(I) * ORT(I) 100 CONTINUE C G = -SIGN(SQRT(H),ORT(M)) H = H - ORT(M) * G ORT(M) = ORT(M) - G C .......... FORM (I-(U*UT)/H) * A .......... DO 130 J = M, N F = 0.0E0 C .......... FOR I=IGH STEP -1 UNTIL M DO -- .......... DO 110 II = M, IGH I = MP - II F = F + ORT(I) * A(I,J) 110 CONTINUE C F = F / H C DO 120 I = M, IGH 120 A(I,J) = A(I,J) - F * ORT(I) C 130 CONTINUE C .......... FORM (I-(U*UT)/H)*A*(I-(U*UT)/H) .......... DO 160 I = 1, IGH F = 0.0E0 C .......... FOR J=IGH STEP -1 UNTIL M DO -- .......... DO 140 JJ = M, IGH J = MP - JJ F = F + ORT(J) * A(I,J) 140 CONTINUE C F = F / H C DO 150 J = M, IGH 150 A(I,J) = A(I,J) - F * ORT(J) C 160 CONTINUE C ORT(M) = SCALE * ORT(M) A(M,M-1) = SCALE * G 180 CONTINUE C 200 RETURN END