SGEEVX (3)  Linux Manuals
NAME
sgeevx.f 
SYNOPSIS
Functions/Subroutines
subroutine sgeevx (BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, RCONDV, WORK, LWORK, IWORK, INFO)
SGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
Function/Subroutine Documentation
subroutine sgeevx (characterBALANC, characterJOBVL, characterJOBVR, characterSENSE, integerN, real, dimension( lda, * )A, integerLDA, real, dimension( * )WR, real, dimension( * )WI, real, dimension( ldvl, * )VL, integerLDVL, real, dimension( ldvr, * )VR, integerLDVR, integerILO, integerIHI, real, dimension( * )SCALE, realABNRM, real, dimension( * )RCONDE, real, dimension( * )RCONDV, real, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerINFO)
SGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
Purpose:

SGEEVX computes for an NbyN real nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors. Optionally also, it computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors (ILO, IHI, SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues (RCONDE), and reciprocal condition numbers for the right eigenvectors (RCONDV). The right eigenvector v(j) of A satisfies A * v(j) = lambda(j) * v(j) where lambda(j) is its eigenvalue. The left eigenvector u(j) of A satisfies u(j)**H * A = lambda(j) * u(j)**H where u(j)**H denotes the conjugatetranspose of u(j). The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest component real. Balancing a matrix means permuting the rows and columns to make it more nearly upper triangular, and applying a diagonal similarity transformation D * A * D**(1), where D is a diagonal matrix, to make its rows and columns closer in norm and the condition numbers of its eigenvalues and eigenvectors smaller. 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 4.10.2 of the LAPACK Users' Guide.
Parameters:

BALANC
BALANC is CHARACTER*1 Indicates how the input matrix should be diagonally scaled and/or permuted to improve the conditioning of its eigenvalues. = 'N': Do not diagonally scale or permute; = 'P': Perform permutations to make the matrix more nearly upper triangular. Do not diagonally scale; = 'S': Diagonally scale the matrix, i.e. replace A by D*A*D**(1), where D is a diagonal matrix chosen to make the rows and columns of A more equal in norm. Do not permute; = 'B': Both diagonally scale and permute A. Computed reciprocal condition numbers will be for the matrix after balancing and/or permuting. Permuting does not change condition numbers (in exact arithmetic), but balancing does.
JOBVLJOBVL is CHARACTER*1 = 'N': left eigenvectors of A are not computed; = 'V': left eigenvectors of A are computed. If SENSE = 'E' or 'B', JOBVL must = 'V'.
JOBVRJOBVR is CHARACTER*1 = 'N': right eigenvectors of A are not computed; = 'V': right eigenvectors of A are computed. If SENSE = 'E' or 'B', JOBVR must = 'V'.
SENSESENSE is CHARACTER*1 Determines which reciprocal condition numbers are computed. = 'N': None are computed; = 'E': Computed for eigenvalues only; = 'V': Computed for right eigenvectors only; = 'B': Computed for eigenvalues and right eigenvectors. If SENSE = 'E' or 'B', both left and right eigenvectors must also be computed (JOBVL = 'V' and JOBVR = 'V').
NN is INTEGER The order of the matrix A. N >= 0.
AA is REAL array, dimension (LDA,N) On entry, the NbyN matrix A. On exit, A has been overwritten. If JOBVL = 'V' or JOBVR = 'V', A contains the real Schur form of the balanced version of the input matrix A.
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
WRWR is REAL array, dimension (N)
WIWI is REAL array, dimension (N) WR and WI contain the real and imaginary parts, respectively, of the computed eigenvalues. Complex conjugate pairs of eigenvalues will appear consecutively with the eigenvalue having the positive imaginary part first.
VLVL is 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 JOBVL = 'N', VL is not referenced. If the jth eigenvalue is real, then u(j) = VL(:,j), the jth column of VL. If the jth and (j+1)st 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).
LDVLLDVL is INTEGER The leading dimension of the array VL. LDVL >= 1; if JOBVL = 'V', LDVL >= N.
VRVR is 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 JOBVR = 'N', VR is not referenced. If the jth eigenvalue is real, then v(j) = VR(:,j), the jth column of VR. If the jth and (j+1)st 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).
LDVRLDVR is INTEGER The leading dimension of the array VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N.
ILOILO is INTEGER
IHIIHI is INTEGER ILO and IHI are integer values determined when A was balanced. The balanced A(i,j) = 0 if I > J and J = 1,...,ILO1 or I = IHI+1,...,N.
SCALESCALE is REAL array, dimension (N) Details of the permutations and scaling factors applied when balancing A. If P(j) is the index of the row and column interchanged with row and column j, and D(j) is the scaling factor applied to row and column j, then SCALE(J) = P(J), for J = 1,...,ILO1 = D(J), for J = ILO,...,IHI = P(J) for J = IHI+1,...,N. The order in which the interchanges are made is N to IHI+1, then 1 to ILO1.
ABNRMABNRM is REAL The onenorm of the balanced matrix (the maximum of the sum of absolute values of elements of any column).
RCONDERCONDE is REAL array, dimension (N) RCONDE(j) is the reciprocal condition number of the jth eigenvalue.
RCONDVRCONDV is REAL array, dimension (N) RCONDV(j) is the reciprocal condition number of the jth right eigenvector.
WORKWORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORKLWORK is INTEGER The dimension of the array WORK. If SENSE = 'N' or 'E', LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V', LWORK >= 3*N. If SENSE = 'V' or 'B', LWORK >= N*(N+6). For good performance, LWORK must generally be larger. 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.
IWORKIWORK is INTEGER array, dimension (2*N2) If SENSE = 'N' or 'E', not referenced.
INFOINFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value. > 0: if INFO = i, the QR algorithm failed to compute all the eigenvalues, and no eigenvectors or condition numbers have been computed; elements 1:ILO1 and i+1:N of WR and WI contain eigenvalues which have converged.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 September 2012
Definition at line 302 of file sgeevx.f.
Author
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