MAGMA  1.6.1
Matrix Algebra for GPU and Multicore Architectures
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Functions

magma_int_t magma_dgehrd (magma_int_t n, magma_int_t ilo, magma_int_t ihi, double *A, magma_int_t lda, double *tau, double *work, magma_int_t lwork, magmaDouble_ptr dT, magma_int_t *info)
 DGEHRD reduces a DOUBLE_PRECISION general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q' * A * Q = H . More...
 
magma_int_t magma_dgehrd2 (magma_int_t n, magma_int_t ilo, magma_int_t ihi, double *A, magma_int_t lda, double *tau, double *work, magma_int_t lwork, magma_int_t *info)
 DGEHRD2 reduces a DOUBLE_PRECISION general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q' * A * Q = H . More...
 
magma_int_t magma_dgehrd_m (magma_int_t n, magma_int_t ilo, magma_int_t ihi, double *A, magma_int_t lda, double *tau, double *work, magma_int_t lwork, double *T, magma_int_t *info)
 DGEHRD reduces a DOUBLE_PRECISION general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q' * A * Q = H . More...
 
magma_int_t magma_dorghr (magma_int_t n, magma_int_t ilo, magma_int_t ihi, double *A, magma_int_t lda, double *tau, magmaDouble_ptr dT, magma_int_t nb, magma_int_t *info)
 DORGHR generates a DOUBLE_PRECISION unitary matrix Q which is defined as the product of IHI-ILO elementary reflectors of order N, as returned by DGEHRD: More...
 
magma_int_t magma_dorghr_m (magma_int_t n, magma_int_t ilo, magma_int_t ihi, double *A, magma_int_t lda, double *tau, double *T, magma_int_t nb, magma_int_t *info)
 DORGHR generates a DOUBLE_PRECISION unitary matrix Q which is defined as the product of IHI-ILO elementary reflectors of order N, as returned by DGEHRD: More...
 
magma_int_t magma_dtrevc3 (magma_side_t side, magma_vec_t howmany, magma_int_t *select, magma_int_t n, double *T, magma_int_t ldt, double *VL, magma_int_t ldvl, double *VR, magma_int_t ldvr, magma_int_t mm, magma_int_t *mout, double *work, magma_int_t lwork, magma_int_t *info)
 DTREVC3 computes some or all of the right and/or left eigenvectors of a real upper quasi-triangular matrix T. More...
 
magma_int_t magma_dtrevc3_mt (magma_side_t side, magma_vec_t howmany, magma_int_t *select, magma_int_t n, double *T, magma_int_t ldt, double *VL, magma_int_t ldvl, double *VR, magma_int_t ldvr, magma_int_t mm, magma_int_t *mout, double *work, magma_int_t lwork, magma_int_t *info)
 DTREVC3_MT computes some or all of the right and/or left eigenvectors of a real upper quasi-triangular matrix T. More...
 
subroutine dtrevc3 (SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM, M, WORK, LWORK, INFO)
 DTREVC3 More...
 

Detailed Description

Function Documentation

subroutine dtrevc3 ( character  SIDE,
character  HOWMNY,
logical, dimension( * )  SELECT,
integer  N,
double precision, dimension( ldt, * )  T,
integer  LDT,
double precision, dimension( ldvl, * )  VL,
integer  LDVL,
double precision, dimension( ldvr, * )  VR,
integer  LDVR,
integer  MM,
integer  M,
double precision, dimension( * )  WORK,
integer  LWORK,
integer  INFO 
)

DTREVC3

Purpose:
 DTREVC3 computes some or all of the right and/or left eigenvectors of
 a real upper quasi-triangular matrix T.
 Matrices of this type are produced by the Schur factorization of
 a real general matrix:  A = Q*T*Q**T, as computed by DHSEQR.

 The right eigenvector x and the left eigenvector y of T corresponding
 to an eigenvalue w are defined by:

    T*x = w*x,     (y**T)*T = w*(y**T)

 where y**T denotes the transpose of the vector y.
 The eigenvalues are not input to this routine, but are read directly
 from the diagonal blocks of T.

 This routine returns the matrices X and/or Y of right and left
 eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
 input matrix. If Q is the orthogonal factor that reduces a matrix
 A to Schur form T, then Q*X and Q*Y are the matrices of right and
 left eigenvectors of A.

 This uses a Level 3 BLAS version of the back transformation.
Parameters
[in]SIDE
          SIDE is CHARACTER*1
          = 'R':  compute right eigenvectors only;
          = 'L':  compute left eigenvectors only;
          = 'B':  compute both right and left eigenvectors.
[in]HOWMNY
          HOWMNY is CHARACTER*1
          = 'A':  compute all right and/or left eigenvectors;
          = 'B':  compute all right and/or left eigenvectors,
                  backtransformed by the matrices in VR and/or VL;
          = 'S':  compute selected right and/or left eigenvectors,
                  as indicated by the logical array SELECT.
[in,out]SELECT
          SELECT is LOGICAL array, dimension (N)
          If HOWMNY = 'S', SELECT specifies the eigenvectors to be
          computed.
          If w(j) is a real eigenvalue, the corresponding real
          eigenvector is computed if SELECT(j) is .TRUE..
          If w(j) and w(j+1) are the real and imaginary parts of a
          complex eigenvalue, the corresponding complex eigenvector is
          computed if either SELECT(j) or SELECT(j+1) is .TRUE., and
          on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to
          .FALSE..
          Not referenced if HOWMNY = 'A' or 'B'.
[in]N
          N is INTEGER
          The order of the matrix T. N >= 0.
[in]T
          T is DOUBLE PRECISION array, dimension (LDT,N)
          The upper quasi-triangular matrix T in Schur canonical form.
[in]LDT
          LDT is INTEGER
          The leading dimension of the array T. LDT >= max(1,N).
[in,out]VL
          VL is DOUBLE PRECISION array, dimension (LDVL,MM)
          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
          contain an N-by-N matrix Q (usually the orthogonal matrix Q
          of Schur vectors returned by DHSEQR).
          On exit, if SIDE = 'L' or 'B', VL contains:
          if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
          if HOWMNY = 'B', the matrix Q*Y;
          if HOWMNY = 'S', the left eigenvectors of T specified by
                           SELECT, stored consecutively in the columns
                           of VL, in the same order as their
                           eigenvalues.
          A complex eigenvector corresponding to a complex eigenvalue
          is stored in two consecutive columns, the first holding the
          real part, and the second the imaginary part.
          Not referenced if SIDE = 'R'.
[in]LDVL
          LDVL is INTEGER
          The leading dimension of the array VL.
          LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N.
[in,out]VR
          VR is DOUBLE PRECISION array, dimension (LDVR,MM)
          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
          contain an N-by-N matrix Q (usually the orthogonal matrix Q
          of Schur vectors returned by DHSEQR).
          On exit, if SIDE = 'R' or 'B', VR contains:
          if HOWMNY = 'A', the matrix X of right eigenvectors of T;
          if HOWMNY = 'B', the matrix Q*X;
          if HOWMNY = 'S', the right eigenvectors of T specified by
                           SELECT, stored consecutively in the columns
                           of VR, in the same order as their
                           eigenvalues.
          A complex eigenvector corresponding to a complex eigenvalue
          is stored in two consecutive columns, the first holding the
          real part and the second the imaginary part.
          Not referenced if SIDE = 'L'.
[in]LDVR
          LDVR is INTEGER
          The leading dimension of the array VR.
          LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N.
[in]MM
          MM is INTEGER
          The number of columns in the arrays VL and/or VR. MM >= M.
[out]M
          M is INTEGER
          The number of columns in the arrays VL and/or VR actually
          used to store the eigenvectors.
          If HOWMNY = 'A' or 'B', M is set to N.
          Each selected real eigenvector occupies one column and each
          selected complex eigenvector occupies two columns.
[out]WORK
          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
[in]LWORK
          LWORK is INTEGER
          The dimension of array WORK. LWORK >= max(1,3*N).
          For optimum performance, LWORK >= N + 2*N*NB, where NB is
          the optimal blocksize.
[out]INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
January 2015
Further Details:
  The algorithm used in this program is basically backward (forward)
  substitution, with scaling to make the the code robust against
  possible overflow.

  Each eigenvector is normalized so that the element of largest
  magnitude has magnitude 1; here the magnitude of a complex number
  (x,y) is taken to be |x| + |y|.
magma_int_t magma_dgehrd ( magma_int_t  n,
magma_int_t  ilo,
magma_int_t  ihi,
double *  A,
magma_int_t  lda,
double *  tau,
double *  work,
magma_int_t  lwork,
magmaDouble_ptr  dT,
magma_int_t *  info 
)

DGEHRD reduces a DOUBLE_PRECISION general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q' * A * Q = H .

This version stores the triangular matrices used in the factorization so that they can be applied directly (i.e., without being recomputed) later. As a result, the application of Q is much faster.

Parameters
[in]nINTEGER The order of the matrix A. N >= 0.
[in]iloINTEGER
[in]ihiINTEGER 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 DGEBAL; otherwise they should be set to 1 and N respectively. See Further Details. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
[in,out]ADOUBLE_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 elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.
[in]ldaINTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]tauDOUBLE_PRECISION array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to zero.
[out]work(workspace) DOUBLE_PRECISION array, dimension (LWORK) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]lworkINTEGER The length of the array WORK. LWORK >= max(1,N). For optimum performance LWORK >= N*NB, where NB is the optimal blocksize.
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.
[out]dTDOUBLE_PRECISION array on the GPU, dimension NB*N, where NB is the optimal blocksize. It stores the NB*NB blocks of the triangular T matrices used in the reduction.
[out]infoINTEGER
  • = 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 (ihi-ilo) elementary reflectors

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real vector with v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit in A(i+2:ihi,i), and tau in TAU(i).

The contents of A are illustrated by the following example, with n = 7, ilo = 2 and ihi = 6:

on entry,                        on exit,

( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      h   h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
(                         a )    (                          a )

where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i).

This implementation follows the hybrid algorithm and notations described in

S. Tomov and J. Dongarra, "Accelerating the reduction to upper Hessenberg form through hybrid GPU-based computing," University of Tennessee Computer Science Technical Report, UT-CS-09-642 (also LAPACK Working Note 219), May 24, 2009.

This version stores the T matrices in dT, for later use in magma_dorghr.

magma_int_t magma_dgehrd2 ( magma_int_t  n,
magma_int_t  ilo,
magma_int_t  ihi,
double *  A,
magma_int_t  lda,
double *  tau,
double *  work,
magma_int_t  lwork,
magma_int_t *  info 
)

DGEHRD2 reduces a DOUBLE_PRECISION general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q' * A * Q = H .

Parameters
[in]nINTEGER The order of the matrix A. N >= 0.
[in]iloINTEGER
[in]ihiINTEGER 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 DGEBAL; otherwise they should be set to 1 and N respectively. See Further Details. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
[in,out]ADOUBLE_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 elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.
[in]ldaINTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]tauDOUBLE_PRECISION array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to zero.
[out]work(workspace) DOUBLE_PRECISION array, dimension (LWORK) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]lworkINTEGER The length of the array WORK. LWORK >= max(1,N). For optimum performance LWORK >= N*NB, where NB is the optimal blocksize.
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.
[out]infoINTEGER
  • = 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 (ihi-ilo) elementary reflectors

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real vector with v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit in A(i+2:ihi,i), and tau in TAU(i).

The contents of A are illustrated by the following example, with n = 7, ilo = 2 and ihi = 6:

on entry,                        on exit,

( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      h   h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
(                         a )    (                          a )

where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i).

This implementation follows the hybrid algorithm and notations described in

S. Tomov and J. Dongarra, "Accelerating the reduction to upper Hessenberg form through hybrid GPU-based computing," University of Tennessee Computer Science Technical Report, UT-CS-09-642 (also LAPACK Working Note 219), May 24, 2009.

magma_int_t magma_dgehrd_m ( magma_int_t  n,
magma_int_t  ilo,
magma_int_t  ihi,
double *  A,
magma_int_t  lda,
double *  tau,
double *  work,
magma_int_t  lwork,
double *  T,
magma_int_t *  info 
)

DGEHRD reduces a DOUBLE_PRECISION general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q' * A * Q = H .

This version stores the triangular matrices used in the factorization so that they can be applied directly (i.e., without being recomputed) later. As a result, the application of Q is much faster.

Parameters
[in]nINTEGER The order of the matrix A. N >= 0.
[in]iloINTEGER
[in]ihiINTEGER 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 DGEBAL; otherwise they should be set to 1 and N respectively. See Further Details. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
[in,out]ADOUBLE_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 elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.
[in]ldaINTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]tauDOUBLE_PRECISION array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to zero.
[out]work(workspace) DOUBLE_PRECISION array, dimension (LWORK) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]lworkINTEGER The length of the array WORK. LWORK >= max(1,N). For optimum performance LWORK >= N*NB, where NB is the optimal blocksize.
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.
[out]TDOUBLE_PRECISION array, dimension NB*N, where NB is the optimal blocksize. It stores the NB*NB blocks of the triangular T matrices used in the reduction.
[out]infoINTEGER
  • = 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 (ihi-ilo) elementary reflectors

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real vector with v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit in A(i+2:ihi,i), and tau in TAU(i).

The contents of A are illustrated by the following example, with n = 7, ilo = 2 and ihi = 6:

on entry,                        on exit,

( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      h   h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
(                         a )    (                          a )

where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i).

This implementation follows the hybrid algorithm and notations described in

S. Tomov and J. Dongarra, "Accelerating the reduction to upper Hessenberg form through hybrid GPU-based computing," University of Tennessee Computer Science Technical Report, UT-CS-09-642 (also LAPACK Working Note 219), May 24, 2009.

This version stores the T matrices, for later use in magma_dorghr.

magma_int_t magma_dorghr ( magma_int_t  n,
magma_int_t  ilo,
magma_int_t  ihi,
double *  A,
magma_int_t  lda,
double *  tau,
magmaDouble_ptr  dT,
magma_int_t  nb,
magma_int_t *  info 
)

DORGHR generates a DOUBLE_PRECISION unitary matrix Q which is defined as the product of IHI-ILO elementary reflectors of order N, as returned by DGEHRD:

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Parameters
[in]nINTEGER The order of the matrix Q. N >= 0.
[in]iloINTEGER
[in]ihiINTEGER ILO and IHI must have the same values as in the previous call of DGEHRD. Q is equal to the unit matrix except in the submatrix Q(ilo+1:ihi,ilo+1:ihi). 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
[in,out]ADOUBLE_PRECISION array, dimension (LDA,N) On entry, the vectors which define the elementary reflectors, as returned by DGEHRD. On exit, the N-by-N unitary matrix Q.
[in]ldaINTEGER The leading dimension of the array A. LDA >= max(1,N).
[in]tauDOUBLE_PRECISION array, dimension (N-1) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by DGEHRD.
[in]dTDOUBLE_PRECISION array on the GPU device. DT contains the T matrices used in blocking the elementary reflectors H(i), e.g., this can be the 9th argument of magma_dgehrd.
[in]nbINTEGER This is the block size used in DGEHRD, and correspondingly the size of the T matrices, used in the factorization, and stored in DT.
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
magma_int_t magma_dorghr_m ( magma_int_t  n,
magma_int_t  ilo,
magma_int_t  ihi,
double *  A,
magma_int_t  lda,
double *  tau,
double *  T,
magma_int_t  nb,
magma_int_t *  info 
)

DORGHR generates a DOUBLE_PRECISION unitary matrix Q which is defined as the product of IHI-ILO elementary reflectors of order N, as returned by DGEHRD:

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Parameters
[in]nINTEGER The order of the matrix Q. N >= 0.
[in]iloINTEGER
[in]ihiINTEGER ILO and IHI must have the same values as in the previous call of DGEHRD. Q is equal to the unit matrix except in the submatrix Q(ilo+1:ihi,ilo+1:ihi). 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
[in,out]ADOUBLE_PRECISION array, dimension (LDA,N) On entry, the vectors which define the elementary reflectors, as returned by DGEHRD. On exit, the N-by-N unitary matrix Q.
[in]ldaINTEGER The leading dimension of the array A. LDA >= max(1,N).
[in]tauDOUBLE_PRECISION array, dimension (N-1) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by DGEHRD.
[in]TDOUBLE_PRECISION array on the GPU device. T contains the T matrices used in blocking the elementary reflectors H(i), e.g., this can be the 9th argument of magma_dgehrd.
[in]nbINTEGER This is the block size used in DGEHRD, and correspondingly the size of the T matrices, used in the factorization, and stored in T.
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
magma_int_t magma_dtrevc3 ( magma_side_t  side,
magma_vec_t  howmany,
magma_int_t *  select,
magma_int_t  n,
double *  T,
magma_int_t  ldt,
double *  VL,
magma_int_t  ldvl,
double *  VR,
magma_int_t  ldvr,
magma_int_t  mm,
magma_int_t *  mout,
double *  work,
magma_int_t  lwork,
magma_int_t *  info 
)

DTREVC3 computes some or all of the right and/or left eigenvectors of a real upper quasi-triangular matrix T.

Matrices of this type are produced by the Schur factorization of a real general matrix: A = Q*T*Q**T, as computed by DHSEQR.

The right eigenvector x and the left eigenvector y of T corresponding to an eigenvalue w are defined by:

T*x = w*x, (y**T)*T = w*(y**T)

where y**T denotes the transpose of the vector y. The eigenvalues are not input to this routine, but are read directly from the diagonal blocks of T.

This routine returns the matrices X and/or Y of right and left eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an input matrix. If Q is the orthogonal factor that reduces a matrix A to Schur form T, then Q*X and Q*Y are the matrices of right and left eigenvectors of A.

This uses a Level 3 BLAS version of the back transformation.

Parameters
[in]sidemagma_side_t
  • = MagmaRight: compute right eigenvectors only;
  • = MagmaLeft: compute left eigenvectors only;
  • = MagmaBothSides: compute both right and left eigenvectors.
[in]howmanymagma_vec_t
  • = MagmaAllVec: compute all right and/or left eigenvectors;
  • = MagmaBacktransVec: compute all right and/or left eigenvectors, backtransformed by the matrices in VR and/or VL;
  • = MagmaSomeVec: compute selected right and/or left eigenvectors, as indicated by the logical array select.
[in,out]selectLOGICAL array, dimension (n) If howmany = MagmaSomeVec, select specifies the eigenvectors to be computed. If w(j) is a real eigenvalue, the corresponding real eigenvector is computed if select(j) is true. If w(j) and w(j+1) are the real and imaginary parts of a complex eigenvalue, the corresponding complex eigenvector is computed if either select(j) or select(j+1) is true, and on exit select(j) is set to true and select(j+1) is set to false. Not referenced if howmany = MagmaAllVec or MagmaBacktransVec.
[in]nINTEGER The order of the matrix T. n >= 0.
[in]TDOUBLE PRECISION array, dimension (ldt,n) The upper quasi-triangular matrix T in Schur canonical form.
[in]ldtINTEGER The leading dimension of the array T. ldt >= max(1,n).
[in,out]VLDOUBLE PRECISION array, dimension (ldvl,mm) On entry, if side = MagmaLeft or MagmaBothSides and howmany = MagmaBacktransVec, VL must contain an n-by-n matrix Q (usually the orthogonal matrix Q of Schur vectors returned by DHSEQR). On exit, if side = MagmaLeft or MagmaBothSides, VL contains: if howmany = MagmaAllVec, the matrix Y of left eigenvectors of T; if howmany = MagmaBacktransVec, the matrix Q*Y; if howmany = MagmaSomeVec, the left eigenvectors of T specified by select, stored consecutively in the columns of VL, in the same order as their eigenvalues. A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part, and the second the imaginary part. Not referenced if side = MagmaRight.
[in]ldvlINTEGER The leading dimension of the array VL. ldvl >= 1, and if side = MagmaLeft or MagmaBothSides, ldvl >= n.
[in,out]VRDOUBLE PRECISION array, dimension (ldvr,mm) On entry, if side = MagmaRight or MagmaBothSides and howmany = MagmaBacktransVec, VR must contain an n-by-n matrix Q (usually the orthogonal matrix Q of Schur vectors returned by DHSEQR). On exit, if side = MagmaRight or MagmaBothSides, VR contains: if howmany = MagmaAllVec, the matrix X of right eigenvectors of T; if howmany = MagmaBacktransVec, the matrix Q*X; if howmany = MagmaSomeVec, the right eigenvectors of T specified by select, stored consecutively in the columns of VR, in the same order as their eigenvalues. A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part and the second the imaginary part. Not referenced if side = MagmaLeft.
[in]ldvrINTEGER The leading dimension of the array VR. ldvr >= 1, and if side = MagmaRight or MagmaBothSides, ldvr >= n.
[in]mmINTEGER The number of columns in the arrays VL and/or VR. mm >= mout.
[out]moutINTEGER The number of columns in the arrays VL and/or VR actually used to store the eigenvectors. If howmany = MagmaAllVec or MagmaBacktransVec, mout is set to n. Each selected real eigenvector occupies one column and each selected complex eigenvector occupies two columns.
work(workspace) DOUBLE PRECISION array, dimension (max(1,lwork))
[in]lworkINTEGER The dimension of array work. lwork >= max(1,3*n). For optimum performance, lwork >= (1 + 2*nb)*n, where nb is the optimal blocksize.
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if info = -i, the i-th argument had an illegal value

Further Details

The algorithm used in this program is basically backward (forward) substitution, with scaling to make the the code robust against possible overflow.

Each eigenvector is normalized so that the element of largest magnitude has magnitude 1; here the magnitude of a complex number (x,y) is taken to be |x| + |y|.

magma_int_t magma_dtrevc3_mt ( magma_side_t  side,
magma_vec_t  howmany,
magma_int_t *  select,
magma_int_t  n,
double *  T,
magma_int_t  ldt,
double *  VL,
magma_int_t  ldvl,
double *  VR,
magma_int_t  ldvr,
magma_int_t  mm,
magma_int_t *  mout,
double *  work,
magma_int_t  lwork,
magma_int_t *  info 
)

DTREVC3_MT computes some or all of the right and/or left eigenvectors of a real upper quasi-triangular matrix T.

Matrices of this type are produced by the Schur factorization of a real general matrix: A = Q*T*Q**T, as computed by DHSEQR.

The right eigenvector x and the left eigenvector y of T corresponding to an eigenvalue w are defined by:

T*x = w*x, (y**T)*T = w*(y**T)

where y**T denotes the transpose of the vector y. The eigenvalues are not input to this routine, but are read directly from the diagonal blocks of T.

This routine returns the matrices X and/or Y of right and left eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an input matrix. If Q is the orthogonal factor that reduces a matrix A to Schur form T, then Q*X and Q*Y are the matrices of right and left eigenvectors of A.

This uses a Level 3 BLAS version of the back transformation. This uses a multi-threaded (mt) implementation.

Parameters
[in]sidemagma_side_t
  • = MagmaRight: compute right eigenvectors only;
  • = MagmaLeft: compute left eigenvectors only;
  • = MagmaBothSides: compute both right and left eigenvectors.
[in]howmanymagma_vec_t
  • = MagmaAllVec: compute all right and/or left eigenvectors;
  • = MagmaBacktransVec: compute all right and/or left eigenvectors, backtransformed by the matrices in VR and/or VL;
  • = MagmaSomeVec: compute selected right and/or left eigenvectors, as indicated by the logical array select.
[in,out]selectLOGICAL array, dimension (n) If howmany = MagmaSomeVec, select specifies the eigenvectors to be computed. If w(j) is a real eigenvalue, the corresponding real eigenvector is computed if select(j) is true. If w(j) and w(j+1) are the real and imaginary parts of a complex eigenvalue, the corresponding complex eigenvector is computed if either select(j) or select(j+1) is true, and on exit select(j) is set to true and select(j+1) is set to false. Not referenced if howmany = MagmaAllVec or MagmaBacktransVec.
[in]nINTEGER The order of the matrix T. n >= 0.
[in]TDOUBLE PRECISION array, dimension (ldt,n) The upper quasi-triangular matrix T in Schur canonical form.
[in]ldtINTEGER The leading dimension of the array T. ldt >= max(1,n).
[in,out]VLDOUBLE PRECISION array, dimension (ldvl,mm) On entry, if side = MagmaLeft or MagmaBothSides and howmany = MagmaBacktransVec, VL must contain an n-by-n matrix Q (usually the orthogonal matrix Q of Schur vectors returned by DHSEQR). On exit, if side = MagmaLeft or MagmaBothSides, VL contains: if howmany = MagmaAllVec, the matrix Y of left eigenvectors of T; if howmany = MagmaBacktransVec, the matrix Q*Y; if howmany = MagmaSomeVec, the left eigenvectors of T specified by select, stored consecutively in the columns of VL, in the same order as their eigenvalues. A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part, and the second the imaginary part. Not referenced if side = MagmaRight.
[in]ldvlINTEGER The leading dimension of the array VL. ldvl >= 1, and if side = MagmaLeft or MagmaBothSides, ldvl >= n.
[in,out]VRDOUBLE PRECISION array, dimension (ldvr,mm) On entry, if side = MagmaRight or MagmaBothSides and howmany = MagmaBacktransVec, VR must contain an n-by-n matrix Q (usually the orthogonal matrix Q of Schur vectors returned by DHSEQR). On exit, if side = MagmaRight or MagmaBothSides, VR contains: if howmany = MagmaAllVec, the matrix X of right eigenvectors of T; if howmany = MagmaBacktransVec, the matrix Q*X; if howmany = MagmaSomeVec, the right eigenvectors of T specified by select, stored consecutively in the columns of VR, in the same order as their eigenvalues. A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part and the second the imaginary part. Not referenced if side = MagmaLeft.
[in]ldvrINTEGER The leading dimension of the array VR. ldvr >= 1, and if side = MagmaRight or MagmaBothSides, ldvr >= n.
[in]mmINTEGER The number of columns in the arrays VL and/or VR. mm >= mout.
[out]moutINTEGER The number of columns in the arrays VL and/or VR actually used to store the eigenvectors. If howmany = MagmaAllVec or MagmaBacktransVec, mout is set to n. Each selected real eigenvector occupies one column and each selected complex eigenvector occupies two columns.
work(workspace) DOUBLE PRECISION array, dimension (max(1,lwork))
[in]lworkINTEGER The dimension of array work. lwork >= max(1,3*n). For optimum performance, lwork >= (1 + 2*nb)*n, where nb is the optimal blocksize.
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if info = -i, the i-th argument had an illegal value

Further Details

The algorithm used in this program is basically backward (forward) substitution, with scaling to make the the code robust against possible overflow.

Each eigenvector is normalized so that the element of largest magnitude has magnitude 1; here the magnitude of a complex number (x,y) is taken to be |x| + |y|.