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- *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
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