533 lines
19 KiB
FortranFixed
533 lines
19 KiB
FortranFixed
SUBROUTINE CGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL,
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$ LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE,
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$ RCONDV, WORK, LWORK, RWORK, INFO )
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*
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* -- LAPACK driver routine (version 3.1) --
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* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
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* November 2006
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*
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* .. Scalar Arguments ..
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CHARACTER BALANC, JOBVL, JOBVR, SENSE
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INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
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REAL ABNRM
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* ..
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* .. Array Arguments ..
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REAL RCONDE( * ), RCONDV( * ), RWORK( * ),
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$ SCALE( * )
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COMPLEX A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
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$ W( * ), WORK( * )
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* ..
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*
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* Purpose
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* =======
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*
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* CGEEVX computes for an N-by-N complex nonsymmetric matrix A, the
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* eigenvalues and, optionally, the left and/or right eigenvectors.
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*
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* Optionally also, it computes a balancing transformation to improve
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* the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
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* SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
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* (RCONDE), and reciprocal condition numbers for the right
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* eigenvectors (RCONDV).
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*
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* The right eigenvector v(j) of A satisfies
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* A * v(j) = lambda(j) * v(j)
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* where lambda(j) is its eigenvalue.
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* The left eigenvector u(j) of A satisfies
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* u(j)**H * A = lambda(j) * u(j)**H
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* where u(j)**H denotes the conjugate transpose of u(j).
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*
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* The computed eigenvectors are normalized to have Euclidean norm
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* equal to 1 and largest component real.
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*
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* Balancing a matrix means permuting the rows and columns to make it
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* more nearly upper triangular, and applying a diagonal similarity
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* transformation D * A * D**(-1), where D is a diagonal matrix, to
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* make its rows and columns closer in norm and the condition numbers
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* of its eigenvalues and eigenvectors smaller. The computed
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* reciprocal condition numbers correspond to the balanced matrix.
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* Permuting rows and columns will not change the condition numbers
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* (in exact arithmetic) but diagonal scaling will. For further
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* explanation of balancing, see section 4.10.2 of the LAPACK
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* Users' Guide.
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*
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* Arguments
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* =========
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*
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* BALANC (input) CHARACTER*1
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* Indicates how the input matrix should be diagonally scaled
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* and/or permuted to improve the conditioning of its
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* eigenvalues.
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* = 'N': Do not diagonally scale or permute;
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* = 'P': Perform permutations to make the matrix more nearly
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* upper triangular. Do not diagonally scale;
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* = 'S': Diagonally scale the matrix, ie. replace A by
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* D*A*D**(-1), where D is a diagonal matrix chosen
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* to make the rows and columns of A more equal in
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* norm. Do not permute;
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* = 'B': Both diagonally scale and permute A.
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*
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* Computed reciprocal condition numbers will be for the matrix
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* after balancing and/or permuting. Permuting does not change
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* condition numbers (in exact arithmetic), but balancing does.
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*
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* JOBVL (input) CHARACTER*1
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* = 'N': left eigenvectors of A are not computed;
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* = 'V': left eigenvectors of A are computed.
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* If SENSE = 'E' or 'B', JOBVL must = 'V'.
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*
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* JOBVR (input) CHARACTER*1
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* = 'N': right eigenvectors of A are not computed;
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* = 'V': right eigenvectors of A are computed.
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* If SENSE = 'E' or 'B', JOBVR must = 'V'.
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*
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* SENSE (input) CHARACTER*1
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* Determines which reciprocal condition numbers are computed.
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* = 'N': None are computed;
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* = 'E': Computed for eigenvalues only;
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* = 'V': Computed for right eigenvectors only;
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* = 'B': Computed for eigenvalues and right eigenvectors.
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*
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* If SENSE = 'E' or 'B', both left and right eigenvectors
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* must also be computed (JOBVL = 'V' and JOBVR = 'V').
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*
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* N (input) INTEGER
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* The order of the matrix A. N >= 0.
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*
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* A (input/output) COMPLEX array, dimension (LDA,N)
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* On entry, the N-by-N matrix A.
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* On exit, A has been overwritten. If JOBVL = 'V' or
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* JOBVR = 'V', A contains the Schur form of the balanced
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* version of the matrix A.
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*
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* LDA (input) INTEGER
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* The leading dimension of the array A. LDA >= max(1,N).
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*
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* W (output) COMPLEX array, dimension (N)
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* W contains the computed eigenvalues.
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*
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* VL (output) COMPLEX array, dimension (LDVL,N)
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* If JOBVL = 'V', the left eigenvectors u(j) are stored one
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* after another in the columns of VL, in the same order
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* as their eigenvalues.
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* If JOBVL = 'N', VL is not referenced.
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* u(j) = VL(:,j), the j-th column of VL.
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*
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* LDVL (input) INTEGER
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* The leading dimension of the array VL. LDVL >= 1; if
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* JOBVL = 'V', LDVL >= N.
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*
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* VR (output) COMPLEX array, dimension (LDVR,N)
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* If JOBVR = 'V', the right eigenvectors v(j) are stored one
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* after another in the columns of VR, in the same order
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* as their eigenvalues.
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* If JOBVR = 'N', VR is not referenced.
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* v(j) = VR(:,j), the j-th column of VR.
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*
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* LDVR (input) INTEGER
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* The leading dimension of the array VR. LDVR >= 1; if
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* JOBVR = 'V', LDVR >= N.
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*
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* ILO (output) INTEGER
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* IHI (output) INTEGER
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* ILO and IHI are integer values determined when A was
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* balanced. The balanced A(i,j) = 0 if I > J and
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* J = 1,...,ILO-1 or I = IHI+1,...,N.
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*
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* SCALE (output) REAL array, dimension (N)
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* Details of the permutations and scaling factors applied
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* when balancing A. If P(j) is the index of the row and column
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* interchanged with row and column j, and D(j) is the scaling
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* factor applied to row and column j, then
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* SCALE(J) = P(J), for J = 1,...,ILO-1
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* = D(J), for J = ILO,...,IHI
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* = P(J) for J = IHI+1,...,N.
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* The order in which the interchanges are made is N to IHI+1,
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* then 1 to ILO-1.
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*
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* ABNRM (output) REAL
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* The one-norm of the balanced matrix (the maximum
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* of the sum of absolute values of elements of any column).
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*
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* RCONDE (output) REAL array, dimension (N)
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* RCONDE(j) is the reciprocal condition number of the j-th
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* eigenvalue.
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*
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* RCONDV (output) REAL array, dimension (N)
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* RCONDV(j) is the reciprocal condition number of the j-th
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* right eigenvector.
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*
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* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
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* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
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*
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* LWORK (input) INTEGER
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* The dimension of the array WORK. If SENSE = 'N' or 'E',
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* LWORK >= max(1,2*N), and if SENSE = 'V' or 'B',
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* LWORK >= N*N+2*N.
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* For good performance, LWORK must generally be larger.
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*
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* If LWORK = -1, then a workspace query is assumed; the routine
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* only calculates the optimal size of the WORK array, returns
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* this value as the first entry of the WORK array, and no error
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* message related to LWORK is issued by XERBLA.
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*
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* RWORK (workspace) REAL array, dimension (2*N)
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*
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* INFO (output) INTEGER
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* = 0: successful exit
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* < 0: if INFO = -i, the i-th argument had an illegal value.
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* > 0: if INFO = i, the QR algorithm failed to compute all the
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* eigenvalues, and no eigenvectors or condition numbers
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* have been computed; elements 1:ILO-1 and i+1:N of W
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* contain eigenvalues which have converged.
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*
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* =====================================================================
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*
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* .. Parameters ..
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REAL ZERO, ONE
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PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
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* ..
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* .. Local Scalars ..
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LOGICAL LQUERY, SCALEA, WANTVL, WANTVR, WNTSNB, WNTSNE,
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$ WNTSNN, WNTSNV
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CHARACTER JOB, SIDE
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INTEGER HSWORK, I, ICOND, IERR, ITAU, IWRK, K, MAXWRK,
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$ MINWRK, NOUT
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REAL ANRM, BIGNUM, CSCALE, EPS, SCL, SMLNUM
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COMPLEX TMP
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* ..
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* .. Local Arrays ..
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LOGICAL SELECT( 1 )
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REAL DUM( 1 )
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* ..
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* .. External Subroutines ..
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EXTERNAL CGEBAK, CGEBAL, CGEHRD, CHSEQR, CLACPY, CLASCL,
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$ CSCAL, CSSCAL, CTREVC, CTRSNA, CUNGHR, SLABAD,
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$ SLASCL, XERBLA
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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INTEGER ILAENV, ISAMAX
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REAL CLANGE, SCNRM2, SLAMCH
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EXTERNAL LSAME, ILAENV, ISAMAX, CLANGE, SCNRM2, SLAMCH
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC AIMAG, CMPLX, CONJG, MAX, REAL, SQRT
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* ..
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* .. Executable Statements ..
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*
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* Test the input arguments
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*
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INFO = 0
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LQUERY = ( LWORK.EQ.-1 )
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WANTVL = LSAME( JOBVL, 'V' )
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WANTVR = LSAME( JOBVR, 'V' )
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WNTSNN = LSAME( SENSE, 'N' )
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WNTSNE = LSAME( SENSE, 'E' )
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WNTSNV = LSAME( SENSE, 'V' )
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WNTSNB = LSAME( SENSE, 'B' )
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IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'S' ) .OR.
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$ LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) ) THEN
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INFO = -1
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ELSE IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
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INFO = -2
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ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
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INFO = -3
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ELSE IF( .NOT.( WNTSNN .OR. WNTSNE .OR. WNTSNB .OR. WNTSNV ) .OR.
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$ ( ( WNTSNE .OR. WNTSNB ) .AND. .NOT.( WANTVL .AND.
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$ WANTVR ) ) ) THEN
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INFO = -4
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ELSE IF( N.LT.0 ) THEN
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INFO = -5
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -7
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ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
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INFO = -10
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ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
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INFO = -12
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END IF
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*
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* Compute workspace
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* (Note: Comments in the code beginning "Workspace:" describe the
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* minimal amount of workspace needed at that point in the code,
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* as well as the preferred amount for good performance.
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* CWorkspace refers to complex workspace, and RWorkspace to real
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* workspace. NB refers to the optimal block size for the
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* immediately following subroutine, as returned by ILAENV.
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* HSWORK refers to the workspace preferred by CHSEQR, as
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* calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
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* the worst case.)
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*
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IF( INFO.EQ.0 ) THEN
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IF( N.EQ.0 ) THEN
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MINWRK = 1
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MAXWRK = 1
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ELSE
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MAXWRK = N + N*ILAENV( 1, 'CGEHRD', ' ', N, 1, N, 0 )
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*
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IF( WANTVL ) THEN
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CALL CHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VL, LDVL,
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$ WORK, -1, INFO )
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ELSE IF( WANTVR ) THEN
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CALL CHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VR, LDVR,
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$ WORK, -1, INFO )
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ELSE
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IF( WNTSNN ) THEN
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CALL CHSEQR( 'E', 'N', N, 1, N, A, LDA, W, VR, LDVR,
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$ WORK, -1, INFO )
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ELSE
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CALL CHSEQR( 'S', 'N', N, 1, N, A, LDA, W, VR, LDVR,
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$ WORK, -1, INFO )
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END IF
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END IF
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HSWORK = WORK( 1 )
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*
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IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN
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MINWRK = 2*N
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IF( .NOT.( WNTSNN .OR. WNTSNE ) )
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$ MINWRK = MAX( MINWRK, N*N + 2*N )
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MAXWRK = MAX( MAXWRK, HSWORK )
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IF( .NOT.( WNTSNN .OR. WNTSNE ) )
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$ MAXWRK = MAX( MAXWRK, N*N + 2*N )
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ELSE
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MINWRK = 2*N
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IF( .NOT.( WNTSNN .OR. WNTSNE ) )
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$ MINWRK = MAX( MINWRK, N*N + 2*N )
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MAXWRK = MAX( MAXWRK, HSWORK )
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MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'CUNGHR',
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$ ' ', N, 1, N, -1 ) )
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IF( .NOT.( WNTSNN .OR. WNTSNE ) )
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$ MAXWRK = MAX( MAXWRK, N*N + 2*N )
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MAXWRK = MAX( MAXWRK, 2*N )
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END IF
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MAXWRK = MAX( MAXWRK, MINWRK )
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END IF
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WORK( 1 ) = MAXWRK
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*
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IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
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INFO = -20
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END IF
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END IF
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*
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'CGEEVX', -INFO )
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RETURN
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ELSE IF( LQUERY ) THEN
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RETURN
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END IF
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*
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* Quick return if possible
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*
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IF( N.EQ.0 )
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$ RETURN
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*
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* Get machine constants
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*
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EPS = SLAMCH( 'P' )
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SMLNUM = SLAMCH( 'S' )
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BIGNUM = ONE / SMLNUM
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CALL SLABAD( SMLNUM, BIGNUM )
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SMLNUM = SQRT( SMLNUM ) / EPS
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BIGNUM = ONE / SMLNUM
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*
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* Scale A if max element outside range [SMLNUM,BIGNUM]
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*
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ICOND = 0
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ANRM = CLANGE( 'M', N, N, A, LDA, DUM )
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SCALEA = .FALSE.
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IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
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SCALEA = .TRUE.
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CSCALE = SMLNUM
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ELSE IF( ANRM.GT.BIGNUM ) THEN
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SCALEA = .TRUE.
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CSCALE = BIGNUM
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END IF
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IF( SCALEA )
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$ CALL CLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
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*
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* Balance the matrix and compute ABNRM
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*
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CALL CGEBAL( BALANC, N, A, LDA, ILO, IHI, SCALE, IERR )
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ABNRM = CLANGE( '1', N, N, A, LDA, DUM )
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IF( SCALEA ) THEN
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DUM( 1 ) = ABNRM
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CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR )
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ABNRM = DUM( 1 )
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END IF
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*
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* Reduce to upper Hessenberg form
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* (CWorkspace: need 2*N, prefer N+N*NB)
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* (RWorkspace: none)
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*
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ITAU = 1
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IWRK = ITAU + N
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CALL CGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
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$ LWORK-IWRK+1, IERR )
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*
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IF( WANTVL ) THEN
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*
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* Want left eigenvectors
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* Copy Householder vectors to VL
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*
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SIDE = 'L'
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CALL CLACPY( 'L', N, N, A, LDA, VL, LDVL )
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*
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* Generate unitary matrix in VL
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* (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
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* (RWorkspace: none)
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*
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CALL CUNGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ),
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$ LWORK-IWRK+1, IERR )
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*
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* Perform QR iteration, accumulating Schur vectors in VL
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* (CWorkspace: need 1, prefer HSWORK (see comments) )
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* (RWorkspace: none)
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*
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IWRK = ITAU
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CALL CHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VL, LDVL,
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$ WORK( IWRK ), LWORK-IWRK+1, INFO )
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*
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IF( WANTVR ) THEN
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*
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* Want left and right eigenvectors
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* Copy Schur vectors to VR
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*
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SIDE = 'B'
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CALL CLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
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END IF
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*
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ELSE IF( WANTVR ) THEN
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*
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* Want right eigenvectors
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* Copy Householder vectors to VR
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*
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SIDE = 'R'
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CALL CLACPY( 'L', N, N, A, LDA, VR, LDVR )
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*
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* Generate unitary matrix in VR
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* (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
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* (RWorkspace: none)
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*
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CALL CUNGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ),
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$ LWORK-IWRK+1, IERR )
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*
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* Perform QR iteration, accumulating Schur vectors in VR
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* (CWorkspace: need 1, prefer HSWORK (see comments) )
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* (RWorkspace: none)
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*
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IWRK = ITAU
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CALL CHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VR, LDVR,
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$ WORK( IWRK ), LWORK-IWRK+1, INFO )
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*
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ELSE
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*
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* Compute eigenvalues only
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* If condition numbers desired, compute Schur form
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*
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IF( WNTSNN ) THEN
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JOB = 'E'
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ELSE
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JOB = 'S'
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END IF
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*
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* (CWorkspace: need 1, prefer HSWORK (see comments) )
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* (RWorkspace: none)
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*
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IWRK = ITAU
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CALL CHSEQR( JOB, 'N', N, ILO, IHI, A, LDA, W, VR, LDVR,
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$ WORK( IWRK ), LWORK-IWRK+1, INFO )
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END IF
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*
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* If INFO > 0 from CHSEQR, then quit
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*
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IF( INFO.GT.0 )
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$ GO TO 50
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*
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IF( WANTVL .OR. WANTVR ) THEN
|
|
*
|
|
* Compute left and/or right eigenvectors
|
|
* (CWorkspace: need 2*N)
|
|
* (RWorkspace: need N)
|
|
*
|
|
CALL CTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
|
|
$ N, NOUT, WORK( IWRK ), RWORK, IERR )
|
|
END IF
|
|
*
|
|
* Compute condition numbers if desired
|
|
* (CWorkspace: need N*N+2*N unless SENSE = 'E')
|
|
* (RWorkspace: need 2*N unless SENSE = 'E')
|
|
*
|
|
IF( .NOT.WNTSNN ) THEN
|
|
CALL CTRSNA( SENSE, 'A', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
|
|
$ RCONDE, RCONDV, N, NOUT, WORK( IWRK ), N, RWORK,
|
|
$ ICOND )
|
|
END IF
|
|
*
|
|
IF( WANTVL ) THEN
|
|
*
|
|
* Undo balancing of left eigenvectors
|
|
*
|
|
CALL CGEBAK( BALANC, 'L', N, ILO, IHI, SCALE, N, VL, LDVL,
|
|
$ IERR )
|
|
*
|
|
* Normalize left eigenvectors and make largest component real
|
|
*
|
|
DO 20 I = 1, N
|
|
SCL = ONE / SCNRM2( N, VL( 1, I ), 1 )
|
|
CALL CSSCAL( N, SCL, VL( 1, I ), 1 )
|
|
DO 10 K = 1, N
|
|
RWORK( K ) = REAL( VL( K, I ) )**2 +
|
|
$ AIMAG( VL( K, I ) )**2
|
|
10 CONTINUE
|
|
K = ISAMAX( N, RWORK, 1 )
|
|
TMP = CONJG( VL( K, I ) ) / SQRT( RWORK( K ) )
|
|
CALL CSCAL( N, TMP, VL( 1, I ), 1 )
|
|
VL( K, I ) = CMPLX( REAL( VL( K, I ) ), ZERO )
|
|
20 CONTINUE
|
|
END IF
|
|
*
|
|
IF( WANTVR ) THEN
|
|
*
|
|
* Undo balancing of right eigenvectors
|
|
*
|
|
CALL CGEBAK( BALANC, 'R', N, ILO, IHI, SCALE, N, VR, LDVR,
|
|
$ IERR )
|
|
*
|
|
* Normalize right eigenvectors and make largest component real
|
|
*
|
|
DO 40 I = 1, N
|
|
SCL = ONE / SCNRM2( N, VR( 1, I ), 1 )
|
|
CALL CSSCAL( N, SCL, VR( 1, I ), 1 )
|
|
DO 30 K = 1, N
|
|
RWORK( K ) = REAL( VR( K, I ) )**2 +
|
|
$ AIMAG( VR( K, I ) )**2
|
|
30 CONTINUE
|
|
K = ISAMAX( N, RWORK, 1 )
|
|
TMP = CONJG( VR( K, I ) ) / SQRT( RWORK( K ) )
|
|
CALL CSCAL( N, TMP, VR( 1, I ), 1 )
|
|
VR( K, I ) = CMPLX( REAL( VR( K, I ) ), ZERO )
|
|
40 CONTINUE
|
|
END IF
|
|
*
|
|
* Undo scaling if necessary
|
|
*
|
|
50 CONTINUE
|
|
IF( SCALEA ) THEN
|
|
CALL CLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, W( INFO+1 ),
|
|
$ MAX( N-INFO, 1 ), IERR )
|
|
IF( INFO.EQ.0 ) THEN
|
|
IF( ( WNTSNV .OR. WNTSNB ) .AND. ICOND.EQ.0 )
|
|
$ CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, RCONDV, N,
|
|
$ IERR )
|
|
ELSE
|
|
CALL CLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, W, N, IERR )
|
|
END IF
|
|
END IF
|
|
*
|
|
WORK( 1 ) = MAXWRK
|
|
RETURN
|
|
*
|
|
* End of CGEEVX
|
|
*
|
|
END
|