*DECK ORTBAK SUBROUTINE ORTBAK (NM, LOW, IGH, A, ORT, M, Z) C***BEGIN PROLOGUE ORTBAK C***PURPOSE Form the eigenvectors of a general real matrix from the C eigenvectors of the upper Hessenberg matrix output from C ORTHES. C***LIBRARY SLATEC (EISPACK) C***CATEGORY D4C4 C***TYPE SINGLE PRECISION (ORTBAK-S, CORTB-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 ORTBAK, 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 This subroutine forms the eigenvectors of a REAL GENERAL C matrix by back transforming those of the corresponding C upper Hessenberg matrix determined by ORTHES. C C On INPUT C C NM must be set to the row dimension of the two-dimensional C array parameters, A and Z, as declared in the calling C program dimension statement. NM is an INTEGER variable. 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. C C A contains some information about the orthogonal trans- C formations used in the reduction to Hessenberg form by C ORTHES in its strict lower triangle. A is a two-dimensional C REAL array, dimensioned A(NM,IGH). C C ORT contains further information about the orthogonal trans- C formations used in the reduction by ORTHES. Only elements C LOW through IGH are used. ORT is a one-dimensional REAL C array, dimensioned ORT(IGH). C C M is the number of columns of Z to be back transformed. C M is an INTEGER variable. C C Z contains the real and imaginary parts of the eigenvectors to C be back transformed in its first M columns. Z is a two- C dimensional REAL array, dimensioned Z(NM,M). C C On OUTPUT C C Z contains the real and imaginary parts of the transformed C eigenvectors in its first M columns. C C ORT has been used for temporary storage as is not restored. C C NOTE that ORTBAK preserves vector Euclidean norms. 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 ORTBAK C INTEGER I,J,M,LA,MM,MP,NM,IGH,KP1,LOW,MP1 REAL A(NM,*),ORT(*),Z(NM,*) REAL G C C***FIRST EXECUTABLE STATEMENT ORTBAK IF (M .EQ. 0) GO TO 200 LA = IGH - 1 KP1 = LOW + 1 IF (LA .LT. KP1) GO TO 200 C .......... FOR MP=IGH-1 STEP -1 UNTIL LOW+1 DO -- .......... DO 140 MM = KP1, LA MP = LOW + IGH - MM IF (A(MP,MP-1) .EQ. 0.0E0) GO TO 140 MP1 = MP + 1 C DO 100 I = MP1, IGH 100 ORT(I) = A(I,MP-1) C DO 130 J = 1, M G = 0.0E0 C DO 110 I = MP, IGH 110 G = G + ORT(I) * Z(I,J) C .......... DIVISOR BELOW IS NEGATIVE OF H FORMED IN ORTHES. C DOUBLE DIVISION AVOIDS POSSIBLE UNDERFLOW .......... G = (G / ORT(MP)) / A(MP,MP-1) C DO 120 I = MP, IGH 120 Z(I,J) = Z(I,J) + G * ORT(I) C 130 CONTINUE C 140 CONTINUE C 200 RETURN END