USAGE: info, v = NumRu::Lapack.dggbak( job, side, ilo, ihi, lscale, rscale, v, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DGGBAK( JOB, SIDE, N, ILO, IHI, LSCALE, RSCALE, M, V, LDV, INFO ) * Purpose * ======= * * DGGBAK forms the right or left eigenvectors of a real generalized * eigenvalue problem A*x = lambda*B*x, by backward transformation on * the computed eigenvectors of the balanced pair of matrices output by * DGGBAL. * * Arguments * ========= * * JOB (input) CHARACTER*1 * Specifies the type of backward transformation required: * = 'N': do nothing, return immediately; * = 'P': do backward transformation for permutation only; * = 'S': do backward transformation for scaling only; * = 'B': do backward transformations for both permutation and * scaling. * JOB must be the same as the argument JOB supplied to DGGBAL. * * SIDE (input) CHARACTER*1 * = 'R': V contains right eigenvectors; * = 'L': V contains left eigenvectors. * * N (input) INTEGER * The number of rows of the matrix V. N >= 0. * * ILO (input) INTEGER * IHI (input) INTEGER * The integers ILO and IHI determined by DGGBAL. * 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. * * LSCALE (input) DOUBLE PRECISION array, dimension (N) * Details of the permutations and/or scaling factors applied * to the left side of A and B, as returned by DGGBAL. * * RSCALE (input) DOUBLE PRECISION array, dimension (N) * Details of the permutations and/or scaling factors applied * to the right side of A and B, as returned by DGGBAL. * * M (input) INTEGER * The number of columns of the matrix V. M >= 0. * * V (input/output) DOUBLE PRECISION array, dimension (LDV,M) * On entry, the matrix of right or left eigenvectors to be * transformed, as returned by DTGEVC. * On exit, V is overwritten by the transformed eigenvectors. * * LDV (input) INTEGER * The leading dimension of the matrix V. LDV >= max(1,N). * * INFO (output) INTEGER * = 0: successful exit. * < 0: if INFO = -i, the i-th argument had an illegal value. * * Further Details * =============== * * See R.C. Ward, Balancing the generalized eigenvalue problem, * SIAM J. Sci. Stat. Comp. 2 (1981), 141-152. * * ===================================================================== * * .. Local Scalars .. LOGICAL LEFTV, RIGHTV INTEGER I, K * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL DSCAL, DSWAP, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX * ..go to the page top

USAGE: ilo, ihi, lscale, rscale, info, a, b = NumRu::Lapack.dggbal( job, a, b, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO ) * Purpose * ======= * * DGGBAL balances a pair of general real matrices (A,B). This * involves, first, permuting A and B by similarity transformations to * isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N * elements on the diagonal; and second, applying a diagonal similarity * transformation to rows and columns ILO to IHI to make the rows * and columns as close in norm as possible. Both steps are optional. * * Balancing may reduce the 1-norm of the matrices, and improve the * accuracy of the computed eigenvalues and/or eigenvectors in the * generalized eigenvalue problem A*x = lambda*B*x. * * Arguments * ========= * * JOB (input) CHARACTER*1 * Specifies the operations to be performed on A and B: * = 'N': none: simply set ILO = 1, IHI = N, LSCALE(I) = 1.0 * and RSCALE(I) = 1.0 for i = 1,...,N. * = 'P': permute only; * = 'S': scale only; * = 'B': both permute and scale. * * N (input) INTEGER * The order of the matrices A and B. N >= 0. * * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) * On entry, the input matrix A. * On exit, A is overwritten by the balanced matrix. * If JOB = 'N', A is not referenced. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * B (input/output) DOUBLE PRECISION array, dimension (LDB,N) * On entry, the input matrix B. * On exit, B is overwritten by the balanced matrix. * If JOB = 'N', B is not referenced. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * ILO (output) INTEGER * IHI (output) INTEGER * ILO and IHI are set to integers such that on exit * A(i,j) = 0 and B(i,j) = 0 if i > j and * j = 1,...,ILO-1 or i = IHI+1,...,N. * If JOB = 'N' or 'S', ILO = 1 and IHI = N. * * LSCALE (output) DOUBLE PRECISION array, dimension (N) * Details of the permutations and scaling factors applied * to the left side of A and B. If P(j) is the index of the * row interchanged with row j, and D(j) * is the scaling factor applied to row j, then * LSCALE(j) = P(j) for J = 1,...,ILO-1 * = 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 ILO-1. * * RSCALE (output) DOUBLE PRECISION array, dimension (N) * Details of the permutations and scaling factors applied * to the right side of A and B. If P(j) is the index of the * column interchanged with column j, and D(j) * is the scaling factor applied to column j, then * LSCALE(j) = P(j) for J = 1,...,ILO-1 * = 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 ILO-1. * * WORK (workspace) DOUBLE PRECISION array, dimension (lwork) * lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and * at least 1 when JOB = 'N' or 'P'. * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value. * * Further Details * =============== * * See R.C. WARD, Balancing the generalized eigenvalue problem, * SIAM J. Sci. Stat. Comp. 2 (1981), 141-152. * * ===================================================================== *go to the page top

USAGE: sdim, alphar, alphai, beta, vsl, vsr, work, info, a, b = NumRu::Lapack.dgges( jobvsl, jobvsr, sort, a, b, [:lwork => lwork, :usage => usage, :help => help]){|a,b,c| ... } FORTRAN MANUAL SUBROUTINE DGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, BWORK, INFO ) * Purpose * ======= * * DGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B), * the generalized eigenvalues, the generalized real Schur form (S,T), * optionally, the left and/or right matrices of Schur vectors (VSL and * VSR). This gives the generalized Schur factorization * * (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T ) * * Optionally, it also orders the eigenvalues so that a selected cluster * of eigenvalues appears in the leading diagonal blocks of the upper * quasi-triangular matrix S and the upper triangular matrix T.The * leading columns of VSL and VSR then form an orthonormal basis for the * corresponding left and right eigenspaces (deflating subspaces). * * (If only the generalized eigenvalues are needed, use the driver * DGGEV instead, which is faster.) * * A generalized eigenvalue for a pair of matrices (A,B) is 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 interpretation for beta=0 or both being zero. * * A pair of matrices (S,T) is in generalized real Schur form if T is * upper triangular with non-negative diagonal and S 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 S will be * "standardized" by making the corresponding elements of T have the * form: * [ a 0 ] * [ 0 b ] * * and the pair of corresponding 2-by-2 blocks in S and T will have a * complex conjugate pair of generalized eigenvalues. * * * 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. * * SORT (input) CHARACTER*1 * Specifies whether or not to order the eigenvalues on the * diagonal of the generalized Schur form. * = 'N': Eigenvalues are not ordered; * = 'S': Eigenvalues are ordered (see SELCTG); * * SELCTG (external procedure) LOGICAL FUNCTION of three DOUBLE PRECISION arguments * SELCTG must be declared EXTERNAL in the calling subroutine. * If SORT = 'N', SELCTG is not referenced. * If SORT = 'S', SELCTG is used to select eigenvalues to sort * to the top left of the Schur form. * An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if * SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either * one of a complex conjugate pair of eigenvalues is selected, * then both complex eigenvalues are selected. * * Note that in the ill-conditioned case, a selected complex * eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j), * BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2 * in this case. * * N (input) INTEGER * The order of the matrices A, B, VSL, and VSR. N >= 0. * * A (input/output) DOUBLE PRECISION array, dimension (LDA, N) * On entry, the first of the pair of matrices. * On exit, A has been overwritten by its generalized Schur * form S. * * LDA (input) INTEGER * The leading dimension of A. LDA >= max(1,N). * * B (input/output) DOUBLE PRECISION array, dimension (LDB, N) * On entry, the second of the pair of matrices. * On exit, B has been overwritten by its generalized Schur * form T. * * LDB (input) INTEGER * The leading dimension of B. LDB >= max(1,N). * * SDIM (output) INTEGER * If SORT = 'N', SDIM = 0. * If SORT = 'S', SDIM = number of eigenvalues (after sorting) * for which SELCTG is true. (Complex conjugate pairs for which * SELCTG is true for either eigenvalue count as 2.) * * ALPHAR (output) DOUBLE PRECISION array, dimension (N) * ALPHAI (output) DOUBLE PRECISION array, dimension (N) * BETA (output) DOUBLE PRECISION 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, * and BETA(j),j=1,...,N are the diagonals of the complex Schur * form (S,T) 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 complex 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 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. * 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). * * VSL (output) DOUBLE PRECISION array, dimension (LDVSL,N) * If JOBVSL = 'V', VSL will contain the left Schur vectors. * 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) DOUBLE PRECISION array, dimension (LDVSR,N) * If JOBVSR = 'V', VSR will contain the right Schur vectors. * 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) DOUBLE PRECISION 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. * If N = 0, LWORK >= 1, else LWORK >= 8*N+16. * 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. * * BWORK (workspace) LOGICAL array, dimension (N) * Not referenced if SORT = 'N'. * * 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: =N+1: other than QZ iteration failed in DHGEQZ. * =N+2: after reordering, roundoff changed values of * some complex eigenvalues so that leading * eigenvalues in the Generalized Schur form no * longer satisfy SELCTG=.TRUE. This could also * be caused due to scaling. * =N+3: reordering failed in DTGSEN. * * ===================================================================== *go to the page top

USAGE: sdim, alphar, alphai, beta, vsl, vsr, rconde, rcondv, work, info, a, b = NumRu::Lapack.dggesx( jobvsl, jobvsr, sort, sense, a, b, [:lwork => lwork, :liwork => liwork, :usage => usage, :help => help]){|a,b,c| ... } FORTRAN MANUAL SUBROUTINE DGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK, LIWORK, BWORK, INFO ) * Purpose * ======= * * DGGESX computes for a pair of N-by-N real nonsymmetric matrices * (A,B), the generalized eigenvalues, the real Schur form (S,T), and, * optionally, the left and/or right matrices of Schur vectors (VSL and * VSR). This gives the generalized Schur factorization * * (A,B) = ( (VSL) S (VSR)**T, (VSL) T (VSR)**T ) * * Optionally, it also orders the eigenvalues so that a selected cluster * of eigenvalues appears in the leading diagonal blocks of the upper * quasi-triangular matrix S and the upper triangular matrix T; computes * a reciprocal condition number for the average of the selected * eigenvalues (RCONDE); and computes a reciprocal condition number for * the right and left deflating subspaces corresponding to the selected * eigenvalues (RCONDV). The leading columns of VSL and VSR then form * an orthonormal basis for the corresponding left and right eigenspaces * (deflating subspaces). * * A generalized eigenvalue for a pair of matrices (A,B) is 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 interpretation for beta=0 or for both being zero. * * A pair of matrices (S,T) is in generalized real Schur form if T is * upper triangular with non-negative diagonal and S 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 S will be * "standardized" by making the corresponding elements of T have the * form: * [ a 0 ] * [ 0 b ] * * and the pair of corresponding 2-by-2 blocks in S and T will have a * complex conjugate pair of generalized eigenvalues. * * * 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. * * SORT (input) CHARACTER*1 * Specifies whether or not to order the eigenvalues on the * diagonal of the generalized Schur form. * = 'N': Eigenvalues are not ordered; * = 'S': Eigenvalues are ordered (see SELCTG). * * SELCTG (external procedure) LOGICAL FUNCTION of three DOUBLE PRECISION arguments * SELCTG must be declared EXTERNAL in the calling subroutine. * If SORT = 'N', SELCTG is not referenced. * If SORT = 'S', SELCTG is used to select eigenvalues to sort * to the top left of the Schur form. * An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if * SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either * one of a complex conjugate pair of eigenvalues is selected, * then both complex eigenvalues are selected. * Note that a selected complex eigenvalue may no longer satisfy * SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering, * since ordering may change the value of complex eigenvalues * (especially if the eigenvalue is ill-conditioned), in this * case INFO is set to N+3. * * SENSE (input) CHARACTER*1 * Determines which reciprocal condition numbers are computed. * = 'N' : None are computed; * = 'E' : Computed for average of selected eigenvalues only; * = 'V' : Computed for selected deflating subspaces only; * = 'B' : Computed for both. * If SENSE = 'E', 'V', or 'B', SORT must equal 'S'. * * N (input) INTEGER * The order of the matrices A, B, VSL, and VSR. N >= 0. * * A (input/output) DOUBLE PRECISION array, dimension (LDA, N) * On entry, the first of the pair of matrices. * On exit, A has been overwritten by its generalized Schur * form S. * * LDA (input) INTEGER * The leading dimension of A. LDA >= max(1,N). * * B (input/output) DOUBLE PRECISION array, dimension (LDB, N) * On entry, the second of the pair of matrices. * On exit, B has been overwritten by its generalized Schur * form T. * * LDB (input) INTEGER * The leading dimension of B. LDB >= max(1,N). * * SDIM (output) INTEGER * If SORT = 'N', SDIM = 0. * If SORT = 'S', SDIM = number of eigenvalues (after sorting) * for which SELCTG is true. (Complex conjugate pairs for which * SELCTG is true for either eigenvalue count as 2.) * * ALPHAR (output) DOUBLE PRECISION array, dimension (N) * ALPHAI (output) DOUBLE PRECISION array, dimension (N) * BETA (output) DOUBLE PRECISION 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 * and BETA(j),j=1,...,N are the diagonals of the complex Schur * form (S,T) 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 complex 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 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. * 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). * * VSL (output) DOUBLE PRECISION array, dimension (LDVSL,N) * If JOBVSL = 'V', VSL will contain the left Schur vectors. * 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) DOUBLE PRECISION array, dimension (LDVSR,N) * If JOBVSR = 'V', VSR will contain the right Schur vectors. * Not referenced if JOBVSR = 'N'. * * LDVSR (input) INTEGER * The leading dimension of the matrix VSR. LDVSR >= 1, and * if JOBVSR = 'V', LDVSR >= N. * * RCONDE (output) DOUBLE PRECISION array, dimension ( 2 ) * If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the * reciprocal condition numbers for the average of the selected * eigenvalues. * Not referenced if SENSE = 'N' or 'V'. * * RCONDV (output) DOUBLE PRECISION array, dimension ( 2 ) * If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the * reciprocal condition numbers for the selected deflating * subspaces. * Not referenced if SENSE = 'N' or 'E'. * * WORK (workspace/output) DOUBLE PRECISION 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. * If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B', * LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else * LWORK >= max( 8*N, 6*N+16 ). * Note that 2*SDIM*(N-SDIM) <= N*N/2. * Note also that an error is only returned if * LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B' * this may not be large enough. * * If LWORK = -1, then a workspace query is assumed; the routine * only calculates the bound on the optimal size of the WORK * array and the minimum size of the IWORK array, returns these * values as the first entries of the WORK and IWORK arrays, and * no error message related to LWORK or LIWORK is issued by * XERBLA. * * IWORK (workspace) INTEGER array, dimension (MAX(1,LIWORK)) * On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK. * * LIWORK (input) INTEGER * The dimension of the array IWORK. * If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise * LIWORK >= N+6. * * If LIWORK = -1, then a workspace query is assumed; the * routine only calculates the bound on the optimal size of the * WORK array and the minimum size of the IWORK array, returns * these values as the first entries of the WORK and IWORK * arrays, and no error message related to LWORK or LIWORK is * issued by XERBLA. * * BWORK (workspace) LOGICAL array, dimension (N) * Not referenced if SORT = 'N'. * * 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: =N+1: other than QZ iteration failed in DHGEQZ * =N+2: after reordering, roundoff changed values of * some complex eigenvalues so that leading * eigenvalues in the Generalized Schur form no * longer satisfy SELCTG=.TRUE. This could also * be caused due to scaling. * =N+3: reordering failed in DTGSEN. * * Further Details * =============== * * An approximate (asymptotic) bound on the average absolute error of * the selected eigenvalues is * * EPS * norm((A, B)) / RCONDE( 1 ). * * An approximate (asymptotic) bound on the maximum angular error in * the computed deflating subspaces is * * EPS * norm((A, B)) / RCONDV( 2 ). * * See LAPACK User's Guide, section 4.11 for more information. * * ===================================================================== *go to the page top

USAGE: alphar, alphai, beta, vl, vr, work, info, a, b = NumRu::Lapack.dggev( jobvl, jobvr, a, b, [:lwork => lwork, :usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO ) * Purpose * ======= * * 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). * * * Arguments * ========= * * 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. * * N (input) INTEGER * The order of the matrices A, B, VL, and VR. N >= 0. * * A (input/output) DOUBLE PRECISION array, dimension (LDA, N) * On entry, the matrix A in the pair (A,B). * On exit, A has been overwritten. * * LDA (input) INTEGER * The leading dimension of A. LDA >= max(1,N). * * B (input/output) DOUBLE PRECISION array, dimension (LDB, N) * On entry, the matrix B in the pair (A,B). * On exit, B has been overwritten. * * LDB (input) INTEGER * The leading dimension of B. LDB >= max(1,N). * * ALPHAR (output) DOUBLE PRECISION array, dimension (N) * ALPHAI (output) DOUBLE PRECISION array, dimension (N) * BETA (output) 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 (output) 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 (input) INTEGER * The leading dimension of the matrix VL. LDVL >= 1, and * if JOBVL = 'V', LDVL >= N. * * VR (output) 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 (input) INTEGER * The leading dimension of the matrix VR. LDVR >= 1, and * if JOBVR = 'V', LDVR >= N. * * WORK (workspace/output) DOUBLE PRECISION 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(1,8*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 (output) 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. * * ===================================================================== *go to the page top

USAGE: alphar, alphai, beta, vl, vr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, info, a, b = NumRu::Lapack.dggevx( balanc, jobvl, jobvr, sense, a, b, [:lwork => lwork, :usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DGGEVX( 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 ) * Purpose * ======= * * DGGEVX 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. * * 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 conjugate-transpose 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N) * ALPHAI (output) DOUBLE PRECISION array, dimension (N) * BETA (output) 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 (output) 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 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) 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 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,...,ILO-1 or i = IHI+1,...,N. * If BALANC = 'N' or 'S', ILO = 1 and IHI = N. * * LSCALE (output) DOUBLE PRECISION 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,...,ILO-1 * = 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 ILO-1. * * RSCALE (output) DOUBLE PRECISION 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,...,ILO-1 * = 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 ILO-1. * * ABNRM (output) DOUBLE PRECISION * The one-norm of the balanced matrix A. * * BBNRM (output) DOUBLE PRECISION * The one-norm of the balanced matrix B. * * RCONDE (output) DOUBLE PRECISION 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 j-th columns of VL and VR all correspond * to the j-th eigenpair. * If SENSE = 'N or 'V', RCONDE is not referenced. * * RCONDV (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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' or 'B', 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 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. * * 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 i-th * 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 i-th 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. * * ===================================================================== *go to the page top

USAGE: x, y, work, info, a, b, d = NumRu::Lapack.dggglm( a, b, d, [:lwork => lwork, :usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DGGGLM( N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK, INFO ) * Purpose * ======= * * DGGGLM solves a general Gauss-Markov linear model (GLM) problem: * * minimize || y ||_2 subject to d = A*x + B*y * x * * where A is an N-by-M matrix, B is an N-by-P matrix, and d is a * given N-vector. It is assumed that M <= N <= M+P, and * * rank(A) = M and rank( A B ) = N. * * Under these assumptions, the constrained equation is always * consistent, and there is a unique solution x and a minimal 2-norm * solution y, which is obtained using a generalized QR factorization * of the matrices (A, B) given by * * A = Q*(R), B = Q*T*Z. * (0) * * In particular, if matrix B is square nonsingular, then the problem * GLM is equivalent to the following weighted linear least squares * problem * * minimize || inv(B)*(d-A*x) ||_2 * x * * where inv(B) denotes the inverse of B. * * Arguments * ========= * * N (input) INTEGER * The number of rows of the matrices A and B. N >= 0. * * M (input) INTEGER * The number of columns of the matrix A. 0 <= M <= N. * * P (input) INTEGER * The number of columns of the matrix B. P >= N-M. * * A (input/output) DOUBLE PRECISION array, dimension (LDA,M) * On entry, the N-by-M matrix A. * On exit, the upper triangular part of the array A contains * the M-by-M upper triangular matrix R. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * B (input/output) DOUBLE PRECISION array, dimension (LDB,P) * On entry, the N-by-P matrix B. * On exit, if N <= P, the upper triangle of the subarray * B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T; * if N > P, the elements on and above the (N-P)th subdiagonal * contain the N-by-P upper trapezoidal matrix T. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * D (input/output) DOUBLE PRECISION array, dimension (N) * On entry, D is the left hand side of the GLM equation. * On exit, D is destroyed. * * X (output) DOUBLE PRECISION array, dimension (M) * Y (output) DOUBLE PRECISION array, dimension (P) * On exit, X and Y are the solutions of the GLM problem. * * WORK (workspace/output) DOUBLE PRECISION 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,N+M+P). * For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB, * where NB is an upper bound for the optimal blocksizes for * DGEQRF, SGERQF, DORMQR and SORMRQ. * * 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 (output) INTEGER * = 0: successful exit. * < 0: if INFO = -i, the i-th argument had an illegal value. * = 1: the upper triangular factor R associated with A in the * generalized QR factorization of the pair (A, B) is * singular, so that rank(A) < M; the least squares * solution could not be computed. * = 2: the bottom (N-M) by (N-M) part of the upper trapezoidal * factor T associated with B in the generalized QR * factorization of the pair (A, B) is singular, so that * rank( A B ) < N; the least squares solution could not * be computed. * * =================================================================== *go to the page top

USAGE: info, a, b, q, z = NumRu::Lapack.dgghrd( compq, compz, ilo, ihi, a, b, q, z, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO ) * Purpose * ======= * * DGGHRD reduces a pair of real matrices (A,B) to generalized upper * Hessenberg form using orthogonal transformations, where A is a * general matrix and B is upper triangular. The form of the * generalized eigenvalue problem is * A*x = lambda*B*x, * and B is typically made upper triangular by computing its QR * factorization and moving the orthogonal matrix Q to the left side * of the equation. * * This subroutine simultaneously reduces A to a Hessenberg matrix H: * Q**T*A*Z = H * and transforms B to another upper triangular matrix T: * Q**T*B*Z = T * in order to reduce the problem to its standard form * H*y = lambda*T*y * where y = Z**T*x. * * The orthogonal matrices Q and Z are determined as products of Givens * rotations. They may either be formed explicitly, or they may be * postmultiplied into input matrices Q1 and Z1, so that * * Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T * * Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T * * If Q1 is the orthogonal matrix from the QR factorization of B in the * original equation A*x = lambda*B*x, then DGGHRD reduces the original * problem to generalized Hessenberg form. * * Arguments * ========= * * COMPQ (input) CHARACTER*1 * = 'N': do not compute Q; * = 'I': Q is initialized to the unit matrix, and the * orthogonal matrix Q is returned; * = 'V': Q must contain an orthogonal matrix Q1 on entry, * and the product Q1*Q is returned. * * COMPZ (input) CHARACTER*1 * = 'N': do not compute Z; * = 'I': Z is initialized to the unit matrix, and the * orthogonal matrix Z is returned; * = 'V': Z must contain an orthogonal matrix Z1 on entry, * and the product Z1*Z is returned. * * N (input) INTEGER * The order of the matrices A and B. N >= 0. * * ILO (input) INTEGER * IHI (input) INTEGER * ILO and IHI mark the rows and columns of A which are to be * reduced. It is assumed that A is already upper triangular * in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are * normally set by a previous call to SGGBAL; otherwise they * should be set to 1 and N respectively. * 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. * * A (input/output) DOUBLE PRECISION array, dimension (LDA, N) * On entry, the N-by-N general matrix to be reduced. * On exit, the upper triangle and the first subdiagonal of A * are overwritten with the upper Hessenberg matrix H, and the * rest is set to zero. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * B (input/output) DOUBLE PRECISION array, dimension (LDB, N) * On entry, the N-by-N upper triangular matrix B. * On exit, the upper triangular matrix T = Q**T B Z. The * elements below the diagonal are set to zero. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N) * On entry, if COMPQ = 'V', the orthogonal matrix Q1, * typically from the QR factorization of B. * On exit, if COMPQ='I', the orthogonal matrix Q, and if * COMPQ = 'V', the product Q1*Q. * Not referenced if COMPQ='N'. * * LDQ (input) INTEGER * The leading dimension of the array Q. * LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise. * * Z (input/output) DOUBLE PRECISION array, dimension (LDZ, N) * On entry, if COMPZ = 'V', the orthogonal matrix Z1. * On exit, if COMPZ='I', the orthogonal matrix Z, and if * COMPZ = 'V', the product Z1*Z. * Not referenced if COMPZ='N'. * * LDZ (input) INTEGER * The leading dimension of the array Z. * LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise. * * INFO (output) INTEGER * = 0: successful exit. * < 0: if INFO = -i, the i-th argument had an illegal value. * * Further Details * =============== * * This routine reduces A to Hessenberg and B to triangular form by * an unblocked reduction, as described in _Matrix_Computations_, * by Golub and Van Loan (Johns Hopkins Press.) * * ===================================================================== *go to the page top

USAGE: x, work, info, a, b, c, d = NumRu::Lapack.dgglse( a, b, c, d, [:lwork => lwork, :usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DGGLSE( M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK, INFO ) * Purpose * ======= * * DGGLSE solves the linear equality-constrained least squares (LSE) * problem: * * minimize || c - A*x ||_2 subject to B*x = d * * where A is an M-by-N matrix, B is a P-by-N matrix, c is a given * M-vector, and d is a given P-vector. It is assumed that * P <= N <= M+P, and * * rank(B) = P and rank( (A) ) = N. * ( (B) ) * * These conditions ensure that the LSE problem has a unique solution, * which is obtained using a generalized RQ factorization of the * matrices (B, A) given by * * B = (0 R)*Q, A = Z*T*Q. * * Arguments * ========= * * M (input) INTEGER * The number of rows of the matrix A. M >= 0. * * N (input) INTEGER * The number of columns of the matrices A and B. N >= 0. * * P (input) INTEGER * The number of rows of the matrix B. 0 <= P <= N <= M+P. * * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) * On entry, the M-by-N matrix A. * On exit, the elements on and above the diagonal of the array * contain the min(M,N)-by-N upper trapezoidal matrix T. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,M). * * B (input/output) DOUBLE PRECISION array, dimension (LDB,N) * On entry, the P-by-N matrix B. * On exit, the upper triangle of the subarray B(1:P,N-P+1:N) * contains the P-by-P upper triangular matrix R. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,P). * * C (input/output) DOUBLE PRECISION array, dimension (M) * On entry, C contains the right hand side vector for the * least squares part of the LSE problem. * On exit, the residual sum of squares for the solution * is given by the sum of squares of elements N-P+1 to M of * vector C. * * D (input/output) DOUBLE PRECISION array, dimension (P) * On entry, D contains the right hand side vector for the * constrained equation. * On exit, D is destroyed. * * X (output) DOUBLE PRECISION array, dimension (N) * On exit, X is the solution of the LSE problem. * * WORK (workspace/output) DOUBLE PRECISION 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,M+N+P). * For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB, * where NB is an upper bound for the optimal blocksizes for * DGEQRF, SGERQF, DORMQR and SORMRQ. * * 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 (output) INTEGER * = 0: successful exit. * < 0: if INFO = -i, the i-th argument had an illegal value. * = 1: the upper triangular factor R associated with B in the * generalized RQ factorization of the pair (B, A) is * singular, so that rank(B) < P; the least squares * solution could not be computed. * = 2: the (N-P) by (N-P) part of the upper trapezoidal factor * T associated with A in the generalized RQ factorization * of the pair (B, A) is singular, so that * rank( (A) ) < N; the least squares solution could not * ( (B) ) * be computed. * * ===================================================================== *go to the page top

USAGE: taua, taub, work, info, a, b = NumRu::Lapack.dggqrf( n, a, b, [:lwork => lwork, :usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO ) * Purpose * ======= * * DGGQRF computes a generalized QR factorization of an N-by-M matrix A * and an N-by-P matrix B: * * A = Q*R, B = Q*T*Z, * * where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal * matrix, and R and T assume one of the forms: * * if N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N, * ( 0 ) N-M N M-N * M * * where R11 is upper triangular, and * * if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P, * P-N N ( T21 ) P * P * * where T12 or T21 is upper triangular. * * In particular, if B is square and nonsingular, the GQR factorization * of A and B implicitly gives the QR factorization of inv(B)*A: * * inv(B)*A = Z'*(inv(T)*R) * * where inv(B) denotes the inverse of the matrix B, and Z' denotes the * transpose of the matrix Z. * * Arguments * ========= * * N (input) INTEGER * The number of rows of the matrices A and B. N >= 0. * * M (input) INTEGER * The number of columns of the matrix A. M >= 0. * * P (input) INTEGER * The number of columns of the matrix B. P >= 0. * * A (input/output) DOUBLE PRECISION array, dimension (LDA,M) * On entry, the N-by-M matrix A. * On exit, the elements on and above the diagonal of the array * contain the min(N,M)-by-M upper trapezoidal matrix R (R is * upper triangular if N >= M); the elements below the diagonal, * with the array TAUA, represent the orthogonal matrix Q as a * product of min(N,M) elementary reflectors (see Further * Details). * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * TAUA (output) DOUBLE PRECISION array, dimension (min(N,M)) * The scalar factors of the elementary reflectors which * represent the orthogonal matrix Q (see Further Details). * * B (input/output) DOUBLE PRECISION array, dimension (LDB,P) * On entry, the N-by-P matrix B. * On exit, if N <= P, the upper triangle of the subarray * B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T; * if N > P, the elements on and above the (N-P)-th subdiagonal * contain the N-by-P upper trapezoidal matrix T; the remaining * elements, with the array TAUB, represent the orthogonal * matrix Z as a product of elementary reflectors (see Further * Details). * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * TAUB (output) DOUBLE PRECISION array, dimension (min(N,P)) * The scalar factors of the elementary reflectors which * represent the orthogonal matrix Z (see Further Details). * * WORK (workspace/output) DOUBLE PRECISION 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,N,M,P). * For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3), * where NB1 is the optimal blocksize for the QR factorization * of an N-by-M matrix, NB2 is the optimal blocksize for the * RQ factorization of an N-by-P matrix, and NB3 is the optimal * blocksize for a call of DORMQR. * * 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 (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value. * * Further Details * =============== * * The matrix Q is represented as a product of elementary reflectors * * Q = H(1) H(2) . . . H(k), where k = min(n,m). * * Each H(i) has the form * * H(i) = I - taua * v * v' * * where taua is a real scalar, and v is a real vector with * v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), * and taua in TAUA(i). * To form Q explicitly, use LAPACK subroutine DORGQR. * To use Q to update another matrix, use LAPACK subroutine DORMQR. * * The matrix Z is represented as a product of elementary reflectors * * Z = H(1) H(2) . . . H(k), where k = min(n,p). * * Each H(i) has the form * * H(i) = I - taub * v * v' * * where taub is a real scalar, and v is a real vector with * v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in * B(n-k+i,1:p-k+i-1), and taub in TAUB(i). * To form Z explicitly, use LAPACK subroutine DORGRQ. * To use Z to update another matrix, use LAPACK subroutine DORMRQ. * * ===================================================================== * * .. Local Scalars .. LOGICAL LQUERY INTEGER LOPT, LWKOPT, NB, NB1, NB2, NB3 * .. * .. External Subroutines .. EXTERNAL DGEQRF, DGERQF, DORMQR, XERBLA * .. * .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV * .. * .. Intrinsic Functions .. INTRINSIC INT, MAX, MIN * ..go to the page top

USAGE: taua, taub, work, info, a, b = NumRu::Lapack.dggrqf( m, p, a, b, [:lwork => lwork, :usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO ) * Purpose * ======= * * DGGRQF computes a generalized RQ factorization of an M-by-N matrix A * and a P-by-N matrix B: * * A = R*Q, B = Z*T*Q, * * where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal * matrix, and R and T assume one of the forms: * * if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N, * N-M M ( R21 ) N * N * * where R12 or R21 is upper triangular, and * * if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P, * ( 0 ) P-N P N-P * N * * where T11 is upper triangular. * * In particular, if B is square and nonsingular, the GRQ factorization * of A and B implicitly gives the RQ factorization of A*inv(B): * * A*inv(B) = (R*inv(T))*Z' * * where inv(B) denotes the inverse of the matrix B, and Z' denotes the * transpose of the matrix Z. * * Arguments * ========= * * M (input) INTEGER * The number of rows of the matrix A. M >= 0. * * P (input) INTEGER * The number of rows of the matrix B. P >= 0. * * N (input) INTEGER * The number of columns of the matrices A and B. N >= 0. * * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) * On entry, the M-by-N matrix A. * On exit, if M <= N, the upper triangle of the subarray * A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R; * if M > N, the elements on and above the (M-N)-th subdiagonal * contain the M-by-N upper trapezoidal matrix R; the remaining * elements, with the array TAUA, represent the orthogonal * matrix Q as a product of elementary reflectors (see Further * Details). * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,M). * * TAUA (output) DOUBLE PRECISION array, dimension (min(M,N)) * The scalar factors of the elementary reflectors which * represent the orthogonal matrix Q (see Further Details). * * B (input/output) DOUBLE PRECISION array, dimension (LDB,N) * On entry, the P-by-N matrix B. * On exit, the elements on and above the diagonal of the array * contain the min(P,N)-by-N upper trapezoidal matrix T (T is * upper triangular if P >= N); the elements below the diagonal, * with the array TAUB, represent the orthogonal matrix Z as a * product of elementary reflectors (see Further Details). * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,P). * * TAUB (output) DOUBLE PRECISION array, dimension (min(P,N)) * The scalar factors of the elementary reflectors which * represent the orthogonal matrix Z (see Further Details). * * WORK (workspace/output) DOUBLE PRECISION 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,N,M,P). * For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3), * where NB1 is the optimal blocksize for the RQ factorization * of an M-by-N matrix, NB2 is the optimal blocksize for the * QR factorization of a P-by-N matrix, and NB3 is the optimal * blocksize for a call of DORMRQ. * * 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 (output) INTEGER * = 0: successful exit * < 0: if INF0= -i, the i-th argument had an illegal value. * * Further Details * =============== * * The matrix Q is represented as a product of elementary reflectors * * Q = H(1) H(2) . . . H(k), where k = min(m,n). * * Each H(i) has the form * * H(i) = I - taua * v * v' * * where taua is a real scalar, and v is a real vector with * v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in * A(m-k+i,1:n-k+i-1), and taua in TAUA(i). * To form Q explicitly, use LAPACK subroutine DORGRQ. * To use Q to update another matrix, use LAPACK subroutine DORMRQ. * * The matrix Z is represented as a product of elementary reflectors * * Z = H(1) H(2) . . . H(k), where k = min(p,n). * * Each H(i) has the form * * H(i) = I - taub * v * v' * * where taub is a real scalar, and v is a real vector with * v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i), * and taub in TAUB(i). * To form Z explicitly, use LAPACK subroutine DORGQR. * To use Z to update another matrix, use LAPACK subroutine DORMQR. * * ===================================================================== * * .. Local Scalars .. LOGICAL LQUERY INTEGER LOPT, LWKOPT, NB, NB1, NB2, NB3 * .. * .. External Subroutines .. EXTERNAL DGEQRF, DGERQF, DORMRQ, XERBLA * .. * .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV * .. * .. Intrinsic Functions .. INTRINSIC INT, MAX, MIN * ..go to the page top

USAGE: k, l, alpha, beta, u, v, q, iwork, info, a, b = NumRu::Lapack.dggsvd( jobu, jobv, jobq, a, b, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, IWORK, INFO ) * Purpose * ======= * * DGGSVD computes the generalized singular value decomposition (GSVD) * of an M-by-N real matrix A and P-by-N real matrix B: * * U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ) * * where U, V and Q are orthogonal matrices, and Z' is the transpose * of Z. Let K+L = the effective numerical rank of the matrix (A',B')', * then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and * D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the * following structures, respectively: * * If M-K-L >= 0, * * K L * D1 = K ( I 0 ) * L ( 0 C ) * M-K-L ( 0 0 ) * * K L * D2 = L ( 0 S ) * P-L ( 0 0 ) * * N-K-L K L * ( 0 R ) = K ( 0 R11 R12 ) * L ( 0 0 R22 ) * * where * * C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), * S = diag( BETA(K+1), ... , BETA(K+L) ), * C**2 + S**2 = I. * * R is stored in A(1:K+L,N-K-L+1:N) on exit. * * If M-K-L < 0, * * K M-K K+L-M * D1 = K ( I 0 0 ) * M-K ( 0 C 0 ) * * K M-K K+L-M * D2 = M-K ( 0 S 0 ) * K+L-M ( 0 0 I ) * P-L ( 0 0 0 ) * * N-K-L K M-K K+L-M * ( 0 R ) = K ( 0 R11 R12 R13 ) * M-K ( 0 0 R22 R23 ) * K+L-M ( 0 0 0 R33 ) * * where * * C = diag( ALPHA(K+1), ... , ALPHA(M) ), * S = diag( BETA(K+1), ... , BETA(M) ), * C**2 + S**2 = I. * * (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored * ( 0 R22 R23 ) * in B(M-K+1:L,N+M-K-L+1:N) on exit. * * The routine computes C, S, R, and optionally the orthogonal * transformation matrices U, V and Q. * * In particular, if B is an N-by-N nonsingular matrix, then the GSVD of * A and B implicitly gives the SVD of A*inv(B): * A*inv(B) = U*(D1*inv(D2))*V'. * If ( A',B')' has orthonormal columns, then the GSVD of A and B is * also equal to the CS decomposition of A and B. Furthermore, the GSVD * can be used to derive the solution of the eigenvalue problem: * A'*A x = lambda* B'*B x. * In some literature, the GSVD of A and B is presented in the form * U'*A*X = ( 0 D1 ), V'*B*X = ( 0 D2 ) * where U and V are orthogonal and X is nonsingular, D1 and D2 are * ``diagonal''. The former GSVD form can be converted to the latter * form by taking the nonsingular matrix X as * * X = Q*( I 0 ) * ( 0 inv(R) ). * * Arguments * ========= * * JOBU (input) CHARACTER*1 * = 'U': Orthogonal matrix U is computed; * = 'N': U is not computed. * * JOBV (input) CHARACTER*1 * = 'V': Orthogonal matrix V is computed; * = 'N': V is not computed. * * JOBQ (input) CHARACTER*1 * = 'Q': Orthogonal matrix Q is computed; * = 'N': Q is not computed. * * M (input) INTEGER * The number of rows of the matrix A. M >= 0. * * N (input) INTEGER * The number of columns of the matrices A and B. N >= 0. * * P (input) INTEGER * The number of rows of the matrix B. P >= 0. * * K (output) INTEGER * L (output) INTEGER * On exit, K and L specify the dimension of the subblocks * described in the Purpose section. * K + L = effective numerical rank of (A',B')'. * * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) * On entry, the M-by-N matrix A. * On exit, A contains the triangular matrix R, or part of R. * See Purpose for details. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,M). * * B (input/output) DOUBLE PRECISION array, dimension (LDB,N) * On entry, the P-by-N matrix B. * On exit, B contains the triangular matrix R if M-K-L < 0. * See Purpose for details. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,P). * * ALPHA (output) DOUBLE PRECISION array, dimension (N) * BETA (output) DOUBLE PRECISION array, dimension (N) * On exit, ALPHA and BETA contain the generalized singular * value pairs of A and B; * ALPHA(1:K) = 1, * BETA(1:K) = 0, * and if M-K-L >= 0, * ALPHA(K+1:K+L) = C, * BETA(K+1:K+L) = S, * or if M-K-L < 0, * ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0 * BETA(K+1:M) =S, BETA(M+1:K+L) =1 * and * ALPHA(K+L+1:N) = 0 * BETA(K+L+1:N) = 0 * * U (output) DOUBLE PRECISION array, dimension (LDU,M) * If JOBU = 'U', U contains the M-by-M orthogonal matrix U. * If JOBU = 'N', U is not referenced. * * LDU (input) INTEGER * The leading dimension of the array U. LDU >= max(1,M) if * JOBU = 'U'; LDU >= 1 otherwise. * * V (output) DOUBLE PRECISION array, dimension (LDV,P) * If JOBV = 'V', V contains the P-by-P orthogonal matrix V. * If JOBV = 'N', V is not referenced. * * LDV (input) INTEGER * The leading dimension of the array V. LDV >= max(1,P) if * JOBV = 'V'; LDV >= 1 otherwise. * * Q (output) DOUBLE PRECISION array, dimension (LDQ,N) * If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q. * If JOBQ = 'N', Q is not referenced. * * LDQ (input) INTEGER * The leading dimension of the array Q. LDQ >= max(1,N) if * JOBQ = 'Q'; LDQ >= 1 otherwise. * * WORK (workspace) DOUBLE PRECISION array, * dimension (max(3*N,M,P)+N) * * IWORK (workspace/output) INTEGER array, dimension (N) * On exit, IWORK stores the sorting information. More * precisely, the following loop will sort ALPHA * for I = K+1, min(M,K+L) * swap ALPHA(I) and ALPHA(IWORK(I)) * endfor * such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N). * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value. * > 0: if INFO = 1, the Jacobi-type procedure failed to * converge. For further details, see subroutine DTGSJA. * * Internal Parameters * =================== * * TOLA DOUBLE PRECISION * TOLB DOUBLE PRECISION * TOLA and TOLB are the thresholds to determine the effective * rank of (A',B')'. Generally, they are set to * TOLA = MAX(M,N)*norm(A)*MAZHEPS, * TOLB = MAX(P,N)*norm(B)*MAZHEPS. * The size of TOLA and TOLB may affect the size of backward * errors of the decomposition. * * Further Details * =============== * * 2-96 Based on modifications by * Ming Gu and Huan Ren, Computer Science Division, University of * California at Berkeley, USA * * ===================================================================== * * .. Local Scalars .. LOGICAL WANTQ, WANTU, WANTV INTEGER I, IBND, ISUB, J, NCYCLE DOUBLE PRECISION ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANGE EXTERNAL LSAME, DLAMCH, DLANGE * .. * .. External Subroutines .. EXTERNAL DCOPY, DGGSVP, DTGSJA, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * ..go to the page top

USAGE: k, l, u, v, q, info, a, b = NumRu::Lapack.dggsvp( jobu, jobv, jobq, a, b, tola, tolb, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, TAU, WORK, INFO ) * Purpose * ======= * * DGGSVP computes orthogonal matrices U, V and Q such that * * N-K-L K L * U'*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0; * L ( 0 0 A23 ) * M-K-L ( 0 0 0 ) * * N-K-L K L * = K ( 0 A12 A13 ) if M-K-L < 0; * M-K ( 0 0 A23 ) * * N-K-L K L * V'*B*Q = L ( 0 0 B13 ) * P-L ( 0 0 0 ) * * where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular * upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0, * otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective * numerical rank of the (M+P)-by-N matrix (A',B')'. Z' denotes the * transpose of Z. * * This decomposition is the preprocessing step for computing the * Generalized Singular Value Decomposition (GSVD), see subroutine * DGGSVD. * * Arguments * ========= * * JOBU (input) CHARACTER*1 * = 'U': Orthogonal matrix U is computed; * = 'N': U is not computed. * * JOBV (input) CHARACTER*1 * = 'V': Orthogonal matrix V is computed; * = 'N': V is not computed. * * JOBQ (input) CHARACTER*1 * = 'Q': Orthogonal matrix Q is computed; * = 'N': Q is not computed. * * M (input) INTEGER * The number of rows of the matrix A. M >= 0. * * P (input) INTEGER * The number of rows of the matrix B. P >= 0. * * N (input) INTEGER * The number of columns of the matrices A and B. N >= 0. * * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) * On entry, the M-by-N matrix A. * On exit, A contains the triangular (or trapezoidal) matrix * described in the Purpose section. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,M). * * B (input/output) DOUBLE PRECISION array, dimension (LDB,N) * On entry, the P-by-N matrix B. * On exit, B contains the triangular matrix described in * the Purpose section. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,P). * * TOLA (input) DOUBLE PRECISION * TOLB (input) DOUBLE PRECISION * TOLA and TOLB are the thresholds to determine the effective * numerical rank of matrix B and a subblock of A. Generally, * they are set to * TOLA = MAX(M,N)*norm(A)*MAZHEPS, * TOLB = MAX(P,N)*norm(B)*MAZHEPS. * The size of TOLA and TOLB may affect the size of backward * errors of the decomposition. * * K (output) INTEGER * L (output) INTEGER * On exit, K and L specify the dimension of the subblocks * described in Purpose. * K + L = effective numerical rank of (A',B')'. * * U (output) DOUBLE PRECISION array, dimension (LDU,M) * If JOBU = 'U', U contains the orthogonal matrix U. * If JOBU = 'N', U is not referenced. * * LDU (input) INTEGER * The leading dimension of the array U. LDU >= max(1,M) if * JOBU = 'U'; LDU >= 1 otherwise. * * V (output) DOUBLE PRECISION array, dimension (LDV,P) * If JOBV = 'V', V contains the orthogonal matrix V. * If JOBV = 'N', V is not referenced. * * LDV (input) INTEGER * The leading dimension of the array V. LDV >= max(1,P) if * JOBV = 'V'; LDV >= 1 otherwise. * * Q (output) DOUBLE PRECISION array, dimension (LDQ,N) * If JOBQ = 'Q', Q contains the orthogonal matrix Q. * If JOBQ = 'N', Q is not referenced. * * LDQ (input) INTEGER * The leading dimension of the array Q. LDQ >= max(1,N) if * JOBQ = 'Q'; LDQ >= 1 otherwise. * * IWORK (workspace) INTEGER array, dimension (N) * * TAU (workspace) DOUBLE PRECISION array, dimension (N) * * WORK (workspace) DOUBLE PRECISION array, dimension (max(3*N,M,P)) * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value. * * * Further Details * =============== * * The subroutine uses LAPACK subroutine DGEQPF for the QR factorization * with column pivoting to detect the effective numerical rank of the * a matrix. It may be replaced by a better rank determination strategy. * * ===================================================================== *go to the page top

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