ggev - ggev: eig, unblocked

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subroutinecggev(characterjobvl,characterjobvr,integern,complex,dimension(lda,*)a,integerlda,complex,dimension(ldb,*)b,integerldb,complex,dimension(*)alpha,complex,dimension(*)beta,complex,dimension(ldvl,*)vl,integerldvl,complex,dimension(ldvr,*)vr,integerldvr,complex,dimension(*)work,integerlwork,real,dimension(*)rwork,integerinfo)CGGEVcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforGEmatricesPurpose: CGGEV computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors. 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 generalized eigenvector v(j) corresponding to the generalized eigenvalue lambda(j) of (A,B) satisfies A * v(j) = lambda(j) * B * v(j). The left generalized eigenvector u(j) corresponding to the generalized eigenvalues lambda(j) of (A,B) satisfies u(j)**H * A = lambda(j) * u(j)**H * B where u(j)**H is the conjugate-transpose of u(j). ParametersJOBVL JOBVL is CHARACTER*1 = 'N': do not compute the left generalized eigenvectors; = 'V': compute the left generalized eigenvectors. JOBVR JOBVR is CHARACTER*1 = 'N': do not compute the right generalized eigenvectors; = 'V': compute the right generalized eigenvectors. N N is INTEGER The order of the matrices A, B, VL, and VR. N >= 0. A A is COMPLEX array, dimension (LDA, N) On entry, the matrix A in the pair (A,B). On exit, A has been overwritten. LDA LDA is INTEGER The leading dimension of A. LDA >= max(1,N). B B is COMPLEX array, dimension (LDB, N) On entry, the matrix B in the pair (A,B). On exit, B has been overwritten. LDB LDB is INTEGER The leading dimension of B. LDB >= max(1,N). ALPHA ALPHA is COMPLEX array, dimension (N) BETA BETA is COMPLEX array, dimension (N) On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues. Note: the quotients ALPHA(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, ALPHA 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 VL is COMPLEX array, dimension (LDVL,N) If JOBVL = 'V', the left generalized eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. Each eigenvector is scaled so the largest component has abs(real part) + abs(imag. part) = 1. Not referenced if JOBVL = 'N'. LDVL LDVL is INTEGER The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N. VR VR is COMPLEX array, dimension (LDVR,N) If JOBVR = 'V', the right generalized eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues. Each eigenvector is scaled so the largest component has abs(real part) + abs(imag. part) = 1. Not referenced if JOBVR = 'N'. LDVR LDVR is INTEGER The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N. WORK WORK is COMPLEX array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,2*N). 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. RWORK RWORK is REAL array, dimension (8*N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. =1,...,N: The QZ iteration failed. No eigenvectors have been calculated, but ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N. > N: =N+1: other then QZ iteration failed in CHGEQZ, =N+2: error return from CTGEVC. Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. subroutinedggev(characterjobvl,characterjobvr,integern,doubleprecision,dimension(lda,*)a,integerlda,doubleprecision,dimension(ldb,*)b,integerldb,doubleprecision,dimension(*)alphar,doubleprecision,dimension(*)alphai,doubleprecision,dimension(*)beta,doubleprecision,dimension(ldvl,*)vl,integerldvl,doubleprecision,dimension(ldvr,*)vr,integerldvr,doubleprecision,dimension(*)work,integerlwork,integerinfo)DGGEVcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforGEmatricesPurpose: DGGEV computes for a pair of N-by-N real nonsymmetric matrices (A,B) the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors. 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 conjugate-transpose of u(j). ParametersJOBVL JOBVL is CHARACTER*1 = 'N': do not compute the left generalized eigenvectors; = 'V': compute the left generalized eigenvectors. JOBVR JOBVR is CHARACTER*1 = 'N': do not compute the right generalized eigenvectors; = 'V': compute the right generalized eigenvectors. N N is INTEGER The order of the matrices A, B, VL, and VR. N >= 0. A A is DOUBLE PRECISION array, dimension (LDA, N) On entry, the matrix A in the pair (A,B). On exit, A has been overwritten. LDA LDA is INTEGER The leading dimension of A. LDA >= max(1,N). B B is DOUBLE PRECISION array, dimension (LDB, N) On entry, the matrix B in the pair (A,B). On exit, B has been overwritten. LDB LDB is INTEGER The leading dimension of B. LDB >= max(1,N). ALPHAR ALPHAR is DOUBLE PRECISION array, dimension (N) ALPHAI ALPHAI is DOUBLE PRECISION array, dimension (N) BETA BETA is DOUBLE PRECISION 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 j-th eigenvalue is real; if positive, then the j-th 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 VL is DOUBLE PRECISION 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 j-th eigenvalue is real, then u(j) = VL(:,j), the j-th column of VL. If the j-th 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 is scaled so the largest component has abs(real part)+abs(imag. part)=1. Not referenced if JOBVL = 'N'. LDVL LDVL is INTEGER The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N. VR VR is DOUBLE PRECISION 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 j-th eigenvalue is real, then v(j) = VR(:,j), the j-th column of VR. If the j-th 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 is scaled so the largest component has abs(real part)+abs(imag. part)=1. Not referenced if JOBVR = 'N'. LDVR LDVR is INTEGER The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N. WORK WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,8*N). 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. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th 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 DHGEQZ. =N+2: error return from DTGEVC. Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. subroutinesggev(characterjobvl,characterjobvr,integern,real,dimension(lda,*)a,integerlda,real,dimension(ldb,*)b,integerldb,real,dimension(*)alphar,real,dimension(*)alphai,real,dimension(*)beta,real,dimension(ldvl,*)vl,integerldvl,real,dimension(ldvr,*)vr,integerldvr,real,dimension(*)work,integerlwork,integerinfo)SGGEVcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforGEmatricesPurpose: SGGEV computes for a pair of N-by-N real nonsymmetric matrices (A,B) the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors. 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 conjugate-transpose of u(j). ParametersJOBVL JOBVL is CHARACTER*1 = 'N': do not compute the left generalized eigenvectors; = 'V': compute the left generalized eigenvectors. JOBVR JOBVR is CHARACTER*1 = 'N': do not compute the right generalized eigenvectors; = 'V': compute the right generalized eigenvectors. N N is INTEGER The order of the matrices A, B, VL, and VR. N >= 0. A A is REAL array, dimension (LDA, N) On entry, the matrix A in the pair (A,B). On exit, A has been overwritten. LDA LDA is INTEGER The leading dimension of A. LDA >= max(1,N). B B is REAL array, dimension (LDB, N) On entry, the matrix B in the pair (A,B). On exit, B has been overwritten. LDB LDB is INTEGER The leading dimension of B. LDB >= max(1,N). ALPHAR ALPHAR is REAL array, dimension (N) ALPHAI ALPHAI is REAL array, dimension (N) BETA BETA is 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 j-th eigenvalue is real; if positive, then the j-th 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 VL 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 the j-th eigenvalue is real, then u(j) = VL(:,j), the j-th column of VL. If the j-th 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 is scaled so the largest component has abs(real part)+abs(imag. part)=1. Not referenced if JOBVL = 'N'. LDVL LDVL is INTEGER The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N. VR VR 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 the j-th eigenvalue is real, then v(j) = VR(:,j), the j-th column of VR. If the j-th 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 is scaled so the largest component has abs(real part)+abs(imag. part)=1. Not referenced if JOBVR = 'N'. LDVR LDVR is INTEGER The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N. WORK WORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,8*N). 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. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th 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. Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. subroutinezggev(characterjobvl,characterjobvr,integern,complex*16,dimension(lda,*)a,integerlda,complex*16,dimension(ldb,*)b,integerldb,complex*16,dimension(*)alpha,complex*16,dimension(*)beta,complex*16,dimension(ldvl,*)vl,integerldvl,complex*16,dimension(ldvr,*)vr,integerldvr,complex*16,dimension(*)work,integerlwork,doubleprecision,dimension(*)rwork,integerinfo)ZGGEVcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforGEmatricesPurpose: ZGGEV computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors. 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 generalized eigenvector v(j) corresponding to the generalized eigenvalue lambda(j) of (A,B) satisfies A * v(j) = lambda(j) * B * v(j). The left generalized eigenvector u(j) corresponding to the generalized eigenvalues lambda(j) of (A,B) satisfies u(j)**H * A = lambda(j) * u(j)**H * B where u(j)**H is the conjugate-transpose of u(j). ParametersJOBVL JOBVL is CHARACTER*1 = 'N': do not compute the left generalized eigenvectors; = 'V': compute the left generalized eigenvectors. JOBVR JOBVR is CHARACTER*1 = 'N': do not compute the right generalized eigenvectors; = 'V': compute the right generalized eigenvectors. N N is INTEGER The order of the matrices A, B, VL, and VR. N >= 0. A A is COMPLEX*16 array, dimension (LDA, N) On entry, the matrix A in the pair (A,B). On exit, A has been overwritten. LDA LDA is INTEGER The leading dimension of A. LDA >= max(1,N). B B is COMPLEX*16 array, dimension (LDB, N) On entry, the matrix B in the pair (A,B). On exit, B has been overwritten. LDB LDB is INTEGER The leading dimension of B. LDB >= max(1,N). ALPHA ALPHA is COMPLEX*16 array, dimension (N) BETA BETA is COMPLEX*16 array, dimension (N) On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues. Note: the quotients ALPHA(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, ALPHA 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 VL is COMPLEX*16 array, dimension (LDVL,N) If JOBVL = 'V', the left generalized eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. Each eigenvector is scaled so the largest component has abs(real part) + abs(imag. part) = 1. Not referenced if JOBVL = 'N'. LDVL LDVL is INTEGER The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N. VR VR is COMPLEX*16 array, dimension (LDVR,N) If JOBVR = 'V', the right generalized eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues. Each eigenvector is scaled so the largest component has abs(real part) + abs(imag. part) = 1. Not referenced if JOBVR = 'N'. LDVR LDVR is INTEGER The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N. WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,2*N). 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. RWORK RWORK is DOUBLE PRECISION array, dimension (8*N) INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. =1,...,N: The QZ iteration failed. No eigenvectors have been calculated, but ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N. > N: =N+1: other then QZ iteration failed in ZHGEQZ, =N+2: error return from ZTGEVC. Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.

ggev - ggev: eig, unblocked

Functions subroutine cggev (jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info) CGGEVcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforGEmatrices subroutine dggev (jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, work, lwork, info) DGGEVcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforGEmatrices subroutine sggev (jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, work, lwork, info) SGGEVcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforGEmatrices subroutine zggev (jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info) ZGGEVcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforGEmatrices