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SUBROUTINE <a name="DGEEVX.1"></a><a href="dgeevx.f.html#DGEEVX.1">DGEEVX</a>( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI,
$ VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,
$ RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK driver routine (version 3.1) --
</span><span class="comment">*</span><span class="comment"> Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment"> November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Scalar Arguments ..
</span> CHARACTER BALANC, JOBVL, JOBVR, SENSE
INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
DOUBLE PRECISION ABNRM
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), RCONDE( * ), RCONDV( * ),
$ SCALE( * ), VL( LDVL, * ), VR( LDVR, * ),
$ WI( * ), WORK( * ), WR( * )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Purpose
</span><span class="comment">*</span><span class="comment"> =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="DGEEVX.24"></a><a href="dgeevx.f.html#DGEEVX.1">DGEEVX</a> computes for an N-by-N real nonsymmetric matrix A, the
</span><span class="comment">*</span><span class="comment"> eigenvalues and, optionally, the left and/or right eigenvectors.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Optionally also, it computes a balancing transformation to improve
</span><span class="comment">*</span><span class="comment"> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
</span><span class="comment">*</span><span class="comment"> SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
</span><span class="comment">*</span><span class="comment"> (RCONDE), and reciprocal condition numbers for the right
</span><span class="comment">*</span><span class="comment"> eigenvectors (RCONDV).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The right eigenvector v(j) of A satisfies
</span><span class="comment">*</span><span class="comment"> A * v(j) = lambda(j) * v(j)
</span><span class="comment">*</span><span class="comment"> where lambda(j) is its eigenvalue.
</span><span class="comment">*</span><span class="comment"> The left eigenvector u(j) of A satisfies
</span><span class="comment">*</span><span class="comment"> u(j)**H * A = lambda(j) * u(j)**H
</span><span class="comment">*</span><span class="comment"> where u(j)**H denotes the conjugate transpose of u(j).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The computed eigenvectors are normalized to have Euclidean norm
</span><span class="comment">*</span><span class="comment"> equal to 1 and largest component real.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Balancing a matrix means permuting the rows and columns to make it
</span><span class="comment">*</span><span class="comment"> more nearly upper triangular, and applying a diagonal similarity
</span><span class="comment">*</span><span class="comment"> transformation D * A * D**(-1), where D is a diagonal matrix, to
</span><span class="comment">*</span><span class="comment"> make its rows and columns closer in norm and the condition numbers
</span><span class="comment">*</span><span class="comment"> of its eigenvalues and eigenvectors smaller. The computed
</span><span class="comment">*</span><span class="comment"> reciprocal condition numbers correspond to the balanced matrix.
</span><span class="comment">*</span><span class="comment"> Permuting rows and columns will not change the condition numbers
</span><span class="comment">*</span><span class="comment"> (in exact arithmetic) but diagonal scaling will. For further
</span><span class="comment">*</span><span class="comment"> explanation of balancing, see section 4.10.2 of the LAPACK
</span><span class="comment">*</span><span class="comment"> Users' Guide.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Arguments
</span><span class="comment">*</span><span class="comment"> =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> BALANC (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> Indicates how the input matrix should be diagonally scaled
</span><span class="comment">*</span><span class="comment"> and/or permuted to improve the conditioning of its
</span><span class="comment">*</span><span class="comment"> eigenvalues.
</span><span class="comment">*</span><span class="comment"> = 'N': Do not diagonally scale or permute;
</span><span class="comment">*</span><span class="comment"> = 'P': Perform permutations to make the matrix more nearly
</span><span class="comment">*</span><span class="comment"> upper triangular. Do not diagonally scale;
</span><span class="comment">*</span><span class="comment"> = 'S': Diagonally scale the matrix, i.e. replace A by
</span><span class="comment">*</span><span class="comment"> D*A*D**(-1), where D is a diagonal matrix chosen
</span><span class="comment">*</span><span class="comment"> to make the rows and columns of A more equal in
</span><span class="comment">*</span><span class="comment"> norm. Do not permute;
</span><span class="comment">*</span><span class="comment"> = 'B': Both diagonally scale and permute A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Computed reciprocal condition numbers will be for the matrix
</span><span class="comment">*</span><span class="comment"> after balancing and/or permuting. Permuting does not change
</span><span class="comment">*</span><span class="comment"> condition numbers (in exact arithmetic), but balancing does.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> JOBVL (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> = 'N': left eigenvectors of A are not computed;
</span><span class="comment">*</span><span class="comment"> = 'V': left eigenvectors of A are computed.
</span><span class="comment">*</span><span class="comment"> If SENSE = 'E' or 'B', JOBVL must = 'V'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> JOBVR (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> = 'N': right eigenvectors of A are not computed;
</span><span class="comment">*</span><span class="comment"> = 'V': right eigenvectors of A are computed.
</span><span class="comment">*</span><span class="comment"> If SENSE = 'E' or 'B', JOBVR must = 'V'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> SENSE (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> Determines which reciprocal condition numbers are computed.
</span><span class="comment">*</span><span class="comment"> = 'N': None are computed;
</span><span class="comment">*</span><span class="comment"> = 'E': Computed for eigenvalues only;
</span><span class="comment">*</span><span class="comment"> = 'V': Computed for right eigenvectors only;
</span><span class="comment">*</span><span class="comment"> = 'B': Computed for eigenvalues and right eigenvectors.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If SENSE = 'E' or 'B', both left and right eigenvectors
</span><span class="comment">*</span><span class="comment"> must also be computed (JOBVL = 'V' and JOBVR = 'V').
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The order of the matrix A. N >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
</span><span class="comment">*</span><span class="comment"> On entry, the N-by-N matrix A.
</span><span class="comment">*</span><span class="comment"> On exit, A has been overwritten. If JOBVL = 'V' or
</span><span class="comment">*</span><span class="comment"> JOBVR = 'V', A contains the real Schur form of the balanced
</span><span class="comment">*</span><span class="comment"> version of the input matrix A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDA (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array A. LDA >= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> WR (output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment"> WI (output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment"> WR and WI contain the real and imaginary parts,
</span><span class="comment">*</span><span class="comment"> respectively, of the computed eigenvalues. Complex
</span><span class="comment">*</span><span class="comment"> conjugate pairs of eigenvalues will appear consecutively
</span><span class="comment">*</span><span class="comment"> with the eigenvalue having the positive imaginary part
</span><span class="comment">*</span><span class="comment"> first.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> VL (output) DOUBLE PRECISION array, dimension (LDVL,N)
</span><span class="comment">*</span><span class="comment"> If JOBVL = 'V', the left eigenvectors u(j) are stored one
</span><span class="comment">*</span><span class="comment"> after another in the columns of VL, in the same order
</span><span class="comment">*</span><span class="comment"> as their eigenvalues.
</span><span class="comment">*</span><span class="comment"> If JOBVL = 'N', VL is not referenced.
</span><span class="comment">*</span><span class="comment"> If the j-th eigenvalue is real, then u(j) = VL(:,j),
</span><span class="comment">*</span><span class="comment"> the j-th column of VL.
</span><span class="comment">*</span><span class="comment"> If the j-th and (j+1)-st eigenvalues form a complex
</span><span class="comment">*</span><span class="comment"> conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
</span><span class="comment">*</span><span class="comment"> u(j+1) = VL(:,j) - i*VL(:,j+1).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDVL (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array VL. LDVL >= 1; if
</span><span class="comment">*</span><span class="comment"> JOBVL = 'V', LDVL >= N.
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