*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