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110 lines
3.9 KiB
Fortran
110 lines
3.9 KiB
Fortran
*DECK ORTBAK
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SUBROUTINE ORTBAK (NM, LOW, IGH, A, ORT, M, Z)
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C***BEGIN PROLOGUE ORTBAK
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C***PURPOSE Form the eigenvectors of a general real matrix from the
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C eigenvectors of the upper Hessenberg matrix output from
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C ORTHES.
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C***LIBRARY SLATEC (EISPACK)
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C***CATEGORY D4C4
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C***TYPE SINGLE PRECISION (ORTBAK-S, CORTB-C)
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C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK
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C***AUTHOR Smith, B. T., et al.
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C***DESCRIPTION
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C
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C This subroutine is a translation of the ALGOL procedure ORTBAK,
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C NUM. MATH. 12, 349-368(1968) by Martin and Wilkinson.
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C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 339-358(1971).
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C
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C This subroutine forms the eigenvectors of a REAL GENERAL
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C matrix by back transforming those of the corresponding
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C upper Hessenberg matrix determined by ORTHES.
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C
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C On INPUT
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C
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C NM must be set to the row dimension of the two-dimensional
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C array parameters, A and Z, as declared in the calling
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C program dimension statement. NM is an INTEGER variable.
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C
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C LOW and IGH are two INTEGER variables determined by the
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C balancing subroutine BALANC. If BALANC has not been
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C used, set LOW=1 and IGH equal to the order of the matrix.
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C
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C A contains some information about the orthogonal trans-
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C formations used in the reduction to Hessenberg form by
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C ORTHES in its strict lower triangle. A is a two-dimensional
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C REAL array, dimensioned A(NM,IGH).
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C
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C ORT contains further information about the orthogonal trans-
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C formations used in the reduction by ORTHES. Only elements
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C LOW through IGH are used. ORT is a one-dimensional REAL
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C array, dimensioned ORT(IGH).
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C
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C M is the number of columns of Z to be back transformed.
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C M is an INTEGER variable.
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C
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C Z contains the real and imaginary parts of the eigenvectors to
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C be back transformed in its first M columns. Z is a two-
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C dimensional REAL array, dimensioned Z(NM,M).
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C
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C On OUTPUT
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C
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C Z contains the real and imaginary parts of the transformed
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C eigenvectors in its first M columns.
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C
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C ORT has been used for temporary storage as is not restored.
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C
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C NOTE that ORTBAK preserves vector Euclidean norms.
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C
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C Questions and comments should be directed to B. S. Garbow,
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C APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
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C ------------------------------------------------------------------
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C
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C***REFERENCES B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow,
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C Y. Ikebe, V. C. Klema and C. B. Moler, Matrix Eigen-
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C system Routines - EISPACK Guide, Springer-Verlag,
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C 1976.
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C***ROUTINES CALLED (NONE)
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C***REVISION HISTORY (YYMMDD)
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C 760101 DATE WRITTEN
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C 890831 Modified array declarations. (WRB)
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C 890831 REVISION DATE from Version 3.2
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C 891214 Prologue converted to Version 4.0 format. (BAB)
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C 920501 Reformatted the REFERENCES section. (WRB)
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C***END PROLOGUE ORTBAK
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C
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INTEGER I,J,M,LA,MM,MP,NM,IGH,KP1,LOW,MP1
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REAL A(NM,*),ORT(*),Z(NM,*)
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REAL G
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C
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C***FIRST EXECUTABLE STATEMENT ORTBAK
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IF (M .EQ. 0) GO TO 200
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LA = IGH - 1
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KP1 = LOW + 1
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IF (LA .LT. KP1) GO TO 200
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C .......... FOR MP=IGH-1 STEP -1 UNTIL LOW+1 DO -- ..........
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DO 140 MM = KP1, LA
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MP = LOW + IGH - MM
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IF (A(MP,MP-1) .EQ. 0.0E0) GO TO 140
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MP1 = MP + 1
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C
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DO 100 I = MP1, IGH
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100 ORT(I) = A(I,MP-1)
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C
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DO 130 J = 1, M
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G = 0.0E0
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C
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DO 110 I = MP, IGH
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110 G = G + ORT(I) * Z(I,J)
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C .......... DIVISOR BELOW IS NEGATIVE OF H FORMED IN ORTHES.
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C DOUBLE DIVISION AVOIDS POSSIBLE UNDERFLOW ..........
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G = (G / ORT(MP)) / A(MP,MP-1)
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C
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DO 120 I = MP, IGH
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120 Z(I,J) = Z(I,J) + G * ORT(I)
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C
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130 CONTINUE
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C
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140 CONTINUE
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C
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200 RETURN
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END
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