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SUBROUTINE <a name="SGGEVX.1"></a><a href="sggevx.f.html#SGGEVX.1">SGGEVX</a>( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
$ IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
$ RCONDV, WORK, LWORK, IWORK, BWORK, 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, LDB, LDVL, LDVR, LWORK, N
REAL ABNRM, BBNRM
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> LOGICAL BWORK( * )
INTEGER IWORK( * )
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), LSCALE( * ),
$ RCONDE( * ), RCONDV( * ), RSCALE( * ),
$ VL( LDVL, * ), VR( LDVR, * ), WORK( * )
<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="SGGEVX.27"></a><a href="sggevx.f.html#SGGEVX.1">SGGEVX</a> computes for a pair of N-by-N real nonsymmetric matrices (A,B)
</span><span class="comment">*</span><span class="comment"> the generalized eigenvalues, and optionally, the left and/or right
</span><span class="comment">*</span><span class="comment"> generalized 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"> LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
</span><span class="comment">*</span><span class="comment"> the eigenvalues (RCONDE), and reciprocal condition numbers for the
</span><span class="comment">*</span><span class="comment"> right eigenvectors (RCONDV).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A generalized eigenvalue for a pair of matrices (A,B) is a scalar
</span><span class="comment">*</span><span class="comment"> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
</span><span class="comment">*</span><span class="comment"> singular. It is usually represented as the pair (alpha,beta), as
</span><span class="comment">*</span><span class="comment"> there is a reasonable interpretation for beta=0, and even for both
</span><span class="comment">*</span><span class="comment"> being zero.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
</span><span class="comment">*</span><span class="comment"> of (A,B) satisfies
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A * v(j) = lambda(j) * B * v(j) .
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
</span><span class="comment">*</span><span class="comment"> of (A,B) satisfies
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> u(j)**H * A = lambda(j) * u(j)**H * B.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where u(j)**H is the conjugate-transpose of u(j).
</span><span class="comment">*</span><span class="comment">
</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"> Specifies the balance option to be performed.
</span><span class="comment">*</span><span class="comment"> = 'N': do not diagonally scale or permute;
</span><span class="comment">*</span><span class="comment"> = 'P': permute only;
</span><span class="comment">*</span><span class="comment"> = 'S': scale only;
</span><span class="comment">*</span><span class="comment"> = 'B': both permute and scale.
</span><span class="comment">*</span><span class="comment"> Computed reciprocal condition numbers will be for the
</span><span class="comment">*</span><span class="comment"> matrices after permuting and/or balancing. Permuting does
</span><span class="comment">*</span><span class="comment"> not change condition numbers (in exact arithmetic), but
</span><span class="comment">*</span><span class="comment"> 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': do not compute the left generalized eigenvectors;
</span><span class="comment">*</span><span class="comment"> = 'V': compute the left generalized eigenvectors.
</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': do not compute the right generalized eigenvectors;
</span><span class="comment">*</span><span class="comment"> = 'V': compute the right generalized eigenvectors.
</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 eigenvectors only;
</span><span class="comment">*</span><span class="comment"> = 'B': computed for eigenvalues and eigenvectors.
</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 matrices A, B, VL, and VR. N >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A (input/output) REAL array, dimension (LDA, N)
</span><span class="comment">*</span><span class="comment"> On entry, the matrix A in the pair (A,B).
</span><span class="comment">*</span><span class="comment"> On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
</span><span class="comment">*</span><span class="comment"> or both, then A contains the first part of the real Schur
</span><span class="comment">*</span><span class="comment"> form of the "balanced" versions of the input A and B.
</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 A. LDA >= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> B (input/output) REAL array, dimension (LDB, N)
</span><span class="comment">*</span><span class="comment"> On entry, the matrix B in the pair (A,B).
</span><span class="comment">*</span><span class="comment"> On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
</span><span class="comment">*</span><span class="comment"> or both, then B contains the second part of the real Schur
</span><span class="comment">*</span><span class="comment"> form of the "balanced" versions of the input A and B.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDB (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of B. LDB >= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ALPHAR (output) REAL array, dimension (N)
</span><span class="comment">*</span><span class="comment"> ALPHAI (output) REAL array, dimension (N)
</span><span class="comment">*</span><span class="comment"> BETA (output) REAL array, dimension (N)
</span><span class="comment">*</span><span class="comment"> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
</span><span class="comment">*</span><span class="comment"> be the generalized eigenvalues. If ALPHAI(j) is zero, then
</span><span class="comment">*</span><span class="comment"> the j-th eigenvalue is real; if positive, then the j-th and
</span><span class="comment">*</span><span class="comment"> (j+1)-st eigenvalues are a complex conjugate pair, with
</span><span class="comment">*</span><span class="comment"> ALPHAI(j+1) negative.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
</span><span class="comment">*</span><span class="comment"> may easily over- or underflow, and BETA(j) may even be zero.
</span><span class="comment">*</span><span class="comment"> Thus, the user should avoid naively computing the ratio
</span><span class="comment">*</span><span class="comment"> ALPHA/BETA. However, ALPHAR and ALPHAI will be always less
</span><span class="comment">*</span><span class="comment"> than and usually comparable with norm(A) in magnitude, and
</span><span class="comment">*</span><span class="comment"> BETA always less than and usually comparable with norm(B).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> VL (output) REAL 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 as
</span><span class="comment">*</span><span class="comment"> their eigenvalues. If the j-th eigenvalue is real, then
</span><span class="comment">*</span><span class="comment"> u(j) = VL(:,j), the j-th column of VL. If the j-th and
</span><span class="comment">*</span><span class="comment"> (j+1)-th eigenvalues form a complex conjugate pair, then
</span><span class="comment">*</span><span class="comment"> u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
</span><span class="comment">*</span><span class="comment"> Each eigenvector will be scaled so the largest component have
</span><span class="comment">*</span><span class="comment"> abs(real part) + abs(imag. part) = 1.
</span><span class="comment">*</span><span class="comment"> Not referenced if JOBVL = 'N'.
</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 matrix VL. LDVL >= 1, and
</span><span class="comment">*</span><span class="comment"> if JOBVL = 'V', LDVL >= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> VR (output) REAL array, dimension (LDVR,N)
</span><span class="comment">*</span><span class="comment"> If JOBVR = 'V', the right eigenvectors v(j) are stored one
</span><span class="comment">*</span><span class="comment"> after another in the columns of VR, in the same order as
</span><span class="comment">*</span><span class="comment"> their eigenvalues. If the j-th eigenvalue is real, then
</span><span class="comment">*</span><span class="comment"> v(j) = VR(:,j), the j-th column of VR. If the j-th and
</span><span class="comment">*</span><span class="comment"> (j+1)-th eigenvalues form a complex conjugate pair, then
</span><span class="comment">*</span><span class="comment"> v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
</span><span class="comment">*</span><span class="comment"> Each eigenvector will be scaled so the largest component have
</span><span class="comment">*</span><span class="comment"> abs(real part) + abs(imag. part) = 1.
</span><span class="comment">*</span><span class="comment"> Not referenced if JOBVR = 'N'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDVR (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the matrix VR. LDVR >= 1, and
</span><span class="comment">*</span><span class="comment"> if JOBVR = 'V', LDVR >= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ILO (output) INTEGER
</span><span class="comment">*</span><span class="comment"> IHI (output) INTEGER
</span><span class="comment">*</span><span class="comment"> ILO and IHI are integer values such that on exit
</span><span class="comment">*</span><span class="comment"> A(i,j) = 0 and B(i,j) = 0 if i > j and
</span><span class="comment">*</span><span class="comment"> j = 1,...,ILO-1 or i = IHI+1,...,N.
</span><span class="comment">*</span><span class="comment"> If BALANC = 'N' or 'S', ILO = 1 and IHI = N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LSCALE (output) REAL array, dimension (N)
</span><span class="comment">*</span><span class="comment"> Details of the permutations and scaling factors applied
</span><span class="comment">*</span><span class="comment"> to the left side of A and B. If PL(j) is the index of the
</span><span class="comment">*</span><span class="comment"> row interchanged with row j, and DL(j) is the scaling
</span><span class="comment">*</span><span class="comment"> factor applied to row j, then
</span><span class="comment">*</span><span class="comment"> LSCALE(j) = PL(j) for j = 1,...,ILO-1
</span><span class="comment">*</span><span class="comment"> = DL(j) for j = ILO,...,IHI
</span><span class="comment">*</span><span class="comment"> = PL(j) for j = IHI+1,...,N.
</span><span class="comment">*</span><span class="comment"> The order in which the interchanges are made is N to IHI+1,
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