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NAME SGEGS - a pair of N-by-N real nonsymmetric matrices A, B SYNOPSIS SUBROUTINE SGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, INFO ) CHARACTER JOBVSL, JOBVSR INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ), WORK( * ) PURPOSE For a pair of N-by-N real nonsymmetric matrices A, B: compute the generalized eigenvalues (alphar +/- alphai*i, beta) compute the real Schur form (A,B) compute the left and/or right Schur vectors (VSL and VSR) The last action is optional -- see the description of JOBVSL and JOBVSR below. (If only the generalized eigenvalues are needed, use the driver SGEGV instead.) A generalized eigenvalue for a pair of matrices (A,B) is, roughly speaking, a scalar w or a ratio alpha/beta = w, such that A - w*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpre- tation for beta=0, and even for both being zero. A good beginning reference is the book, "Matrix Computations", by G. Golub & C. van Loan (Johns Hopkins U. Press) The (generalized) Schur form of a pair of matrices is the result of multiplying both matrices on the left by one orthogonal matrix and both on the right by another orthogo- nal matrix, these two orthogonal matrices being chosen so as to bring the pair of matrices into (real) Schur form. A pair of matrices A, B is in generalized real Schur form if B is upper triangular with non-negative diagonal and A is block upper triangular with 1-by-1 and 2-by-2 blocks. 1- by-1 blocks correspond to real generalized eigenvalues, while 2-by-2 blocks of A will be "standardized" by making the corresponding entries of B have the form: [ a 0 ] [ 0 b ] and the pair of corresponding 2-by-2 blocks in A and B will have a complex conjugate pair of generalized eigenvalues. The left and right Schur vectors are the columns of VSL and VSR, respectively, where VSL and VSR are the orthogonal matrices which reduce A and B to Schur form: Schur form of (A,B) = ( (VSL)**T A (VSR), (VSL)**T B (VSR) ) ARGUMENTS JOBVSL (input) CHARACTER*1 = 'N': do not compute the left Schur vectors; = 'V': compute the left Schur vectors. JOBVSR (input) CHARACTER*1 = 'N': do not compute the right Schur vectors; = 'V': compute the right Schur vectors. N (input) INTEGER The number of rows and columns in the matrices A, B, VSL, and VSR. N >= 0. A (input/output) REAL array, dimension (LDA, N) On entry, the first of the pair of matrices whose generalized eigenvalues and (optionally) Schur vec- tors are to be computed. On exit, the generalized Schur form of A. Note: to avoid overflow, the Fro- benius norm of the matrix A should be less than the overflow threshold. LDA (input) INTEGER The leading dimension of A. LDA >= max(1,N). B (input/output) REAL array, dimension (LDB, N) On entry, the second of the pair of matrices whose generalized eigenvalues and (optionally) Schur vec- tors are to be computed. On exit, the generalized Schur form of B. Note: to avoid overflow, the Fro- benius norm of the matrix B should be less than the overflow threshold. 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. ALPHAR(j) + ALPHAI(j)*i, j=1,...,N and BETA(j),j=1,...,N are the diagonals of the complex Schur form (A,B) that would result if the 2-by-2 diagonal blocks of the real Schur form of (A,B) were further reduced to triangular form using 2-by-2 com- plex unitary transformations. If ALPHAI(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex con- jugate 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. How- ever, 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). VSL (output) REAL array, dimension (LDVSL,N) If JOBVSL = 'V', VSL will contain the left Schur vectors. (See "Purpose", above.) Not referenced if JOBVSL = 'N'. LDVSL (input) INTEGER The leading dimension of the matrix VSL. LDVSL >=1, and if JOBVSL = 'V', LDVSL >= N. VSR (output) REAL array, dimension (LDVSR,N) If JOBVSR = 'V', VSR will contain the right Schur vectors. (See "Purpose", above.) Not referenced if JOBVSR = 'N'. LDVSR (input) INTEGER The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= N. WORK (workspace/output) REAL array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK >= max(1,4*N). For good performance, LWORK must gen- erally be larger. To compute the optimal value of LWORK, call ILAENV to get blocksizes (for SGEQRF, SORMQR, and SORGQR.) Then compute: NB -- MAX of the blocksizes for SGEQRF, SORMQR, and SORGQR The optimal LWORK is 2*N + N*(NB+1). INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. = 1,...,N: The QZ iteration failed. (A,B) are not in Schur form, but ALPHAR(j), ALPHAI(j), and BETA(j) should be correct for j=INFO+1,...,N. > N: errors that usually indicate LAPACK problems: =N+1: error return from SGGBAL =N+2: error return from SGEQRF =N+3: error return from SORMQR =N+4: error return from SORGQR =N+5: error return from SGGHRD =N+6: error return from SHGEQZ (other than failed iteration) =N+7: error return from SGGBAK (computing VSL) =N+8: error return from SGGBAK (computing VSR) =N+9: error return from SLASCL (various places)