SYNOPSIS
 SUBROUTINE SGGEVX(
 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 )
 CHARACTER BALANC, JOBVL, JOBVR, SENSE
 INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
 REAL ABNRM, BBNRM
 LOGICAL BWORK( * )
 INTEGER IWORK( * )
 REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA( * ), LSCALE( * ), RCONDE( * ), RCONDV( * ), RSCALE( * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )
PURPOSE
SGGEVX computes for a pair of NbyN real nonsymmetric matrices (A,B) the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors.Optionally also, it computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors (ILO, IHI, LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for the eigenvalues (RCONDE), and reciprocal condition numbers for the right eigenvectors (RCONDV).
A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda or a ratio alpha/beta = lambda, such that A  lambda*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0, and even for both being zero.
The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies
A * v(j) = lambda(j) * B * v(j) .
The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies
u(j)**H * A = lambda(j) * u(j)**H * B.
where u(j)**H is the conjugatetranspose of u(j).
ARGUMENTS
 BALANC (input) CHARACTER*1

Specifies the balance option to be performed.
= 'N': do not diagonally scale or permute;
= 'P': permute only;
= 'S': scale only;
= 'B': both permute and scale. Computed reciprocal condition numbers will be for the matrices after permuting and/or balancing. Permuting does not change condition numbers (in exact arithmetic), but balancing does.  JOBVL (input) CHARACTER*1

= 'N': do not compute the left generalized eigenvectors;
= 'V': compute the left generalized eigenvectors.  JOBVR (input) CHARACTER*1

= 'N': do not compute the right generalized eigenvectors;
= 'V': compute the right generalized eigenvectors.  SENSE (input) CHARACTER*1

Determines which reciprocal condition numbers are computed.
= 'N': none are computed;
= 'E': computed for eigenvalues only;
= 'V': computed for eigenvectors only;
= 'B': computed for eigenvalues and eigenvectors.  N (input) INTEGER
 The order of the matrices A, B, VL, and VR. N >= 0.
 A (input/output) REAL array, dimension (LDA, N)
 On entry, the matrix A in the pair (A,B). On exit, A has been overwritten. If JOBVL='V' or JOBVR='V' or both, then A contains the first part of the real Schur form of the "balanced" versions of the input A and B.
 LDA (input) INTEGER
 The leading dimension of A. LDA >= max(1,N).
 B (input/output) REAL array, dimension (LDB, N)
 On entry, the matrix B in the pair (A,B). On exit, B has been overwritten. If JOBVL='V' or JOBVR='V' or both, then B contains the second part of the real Schur form of the "balanced" versions of the input A and B.
 LDB (input) INTEGER
 The leading dimension of B. LDB >= max(1,N).
 ALPHAR (output) REAL array, dimension (N)
 ALPHAI (output) REAL array, dimension (N) BETA (output) REAL array, dimension (N) On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigenvalues. If ALPHAI(j) is zero, then the jth eigenvalue is real; if positive, then the jth and (j+1)st eigenvalues are a complex conjugate pair, with ALPHAI(j+1) negative. Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may easily over or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio ALPHA/BETA. However, ALPHAR and ALPHAI will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B).
 VL (output) REAL array, dimension (LDVL,N)
 If JOBVL = 'V', the left eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. If the jth eigenvalue is real, then u(j) = VL(:,j), the jth column of VL. If the jth and (j+1)th eigenvalues form a complex conjugate pair, then u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)i*VL(:,j+1). Each eigenvector will be scaled so the largest component have abs(real part) + abs(imag. part) = 1. Not referenced if JOBVL = 'N'.
 LDVL (input) INTEGER
 The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N.
 VR (output) REAL array, dimension (LDVR,N)
 If JOBVR = 'V', the right eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues. If the jth eigenvalue is real, then v(j) = VR(:,j), the jth column of VR. If the jth and (j+1)th eigenvalues form a complex conjugate pair, then v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)i*VR(:,j+1). Each eigenvector will be scaled so the largest component have abs(real part) + abs(imag. part) = 1. Not referenced if JOBVR = 'N'.
 LDVR (input) INTEGER
 The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N.
 ILO (output) INTEGER
 IHI (output) INTEGER ILO and IHI are integer values such that on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j = 1,...,ILO1 or i = IHI+1,...,N. If BALANC = 'N' or 'S', ILO = 1 and IHI = N.
 LSCALE (output) REAL array, dimension (N)
 Details of the permutations and scaling factors applied to the left side of A and B. If PL(j) is the index of the row interchanged with row j, and DL(j) is the scaling factor applied to row j, then LSCALE(j) = PL(j) for j = 1,...,ILO1 = DL(j) for j = ILO,...,IHI = PL(j) for j = IHI+1,...,N. The order in which the interchanges are made is N to IHI+1, then 1 to ILO1.
 RSCALE (output) REAL array, dimension (N)
 Details of the permutations and scaling factors applied to the right side of A and B. If PR(j) is the index of the column interchanged with column j, and DR(j) is the scaling factor applied to column j, then RSCALE(j) = PR(j) for j = 1,...,ILO1 = DR(j) for j = ILO,...,IHI = PR(j) for j = IHI+1,...,N The order in which the interchanges are made is N to IHI+1, then 1 to ILO1.
 ABNRM (output) REAL
 The onenorm of the balanced matrix A.
 BBNRM (output) REAL
 The onenorm of the balanced matrix B.
 RCONDE (output) REAL array, dimension (N)
 If SENSE = 'E' or 'B', the reciprocal condition numbers of the eigenvalues, stored in consecutive elements of the array. For a complex conjugate pair of eigenvalues two consecutive elements of RCONDE are set to the same value. Thus RCONDE(j), RCONDV(j), and the jth columns of VL and VR all correspond to the jth eigenpair. If SENSE = 'N' or 'V', RCONDE is not referenced.
 RCONDV (output) REAL array, dimension (N)
 If SENSE = 'V' or 'B', the estimated reciprocal condition numbers of the eigenvectors, stored in consecutive elements of the array. For a complex eigenvector two consecutive elements of RCONDV are set to the same value. If the eigenvalues cannot be reordered to compute RCONDV(j), RCONDV(j) is set to 0; this can only occur when the true value would be very small anyway. If SENSE = 'N' or 'E', RCONDV is not referenced.
 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
 LWORK (input) INTEGER
 The dimension of the array WORK. LWORK >= max(1,2*N). If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V', LWORK >= max(1,6*N). If SENSE = 'E', LWORK >= max(1,10*N). If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16. If LWORK = 1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
 IWORK (workspace) INTEGER array, dimension (N+6)
 If SENSE = 'E', IWORK is not referenced.
 BWORK (workspace) LOGICAL array, dimension (N)
 If SENSE = 'N', BWORK is not referenced.
 INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value.
= 1,...,N: The QZ iteration failed. No eigenvectors have been calculated, but ALPHAR(j), ALPHAI(j), and BETA(j) should be correct for j=INFO+1,...,N. > N: =N+1: other than QZ iteration failed in SHGEQZ.
=N+2: error return from STGEVC.
FURTHER DETAILS
Balancing a matrix pair (A,B) includes, first, permuting rows and columns to isolate eigenvalues, second, applying diagonal similarity transformation to the rows and columns to make the rows and columns as close in norm as possible. The computed reciprocal condition numbers correspond to the balanced matrix. Permuting rows and columns will not change the condition numbers (in exact arithmetic) but diagonal scaling will. For further explanation of balancing, see section 4.11.1.2 of LAPACK Users' Guide.An approximate error bound on the chordal distance between the ith computed generalized eigenvalue w and the corresponding exact eigenvalue lambda is
chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I) An approximate error bound for the angle between the ith computed eigenvector VL(i) or VR(i) is given by
EPS * norm(ABNRM, BBNRM) / DIF(i).
For further explanation of the reciprocal condition numbers RCONDE and RCONDV, see section 4.11 of LAPACK User's Guide.