larzt(3)

NAME
larzt - larzt: generate T matrix

SYNOPSIS

  Functions
subroutine clarzt (direct, storev, n, k, v, ldv, tau, t, ldt)
CLARZT forms the triangular factor T of a block reflector H = I - vtvH.
subroutine dlarzt (direct, storev, n, k, v, ldv, tau, t, ldt)
DLARZT forms the triangular factor T of a block reflector H = I - vtvH.
subroutine slarzt (direct, storev, n, k, v, ldv, tau, t, ldt)
SLARZT forms the triangular factor T of a block reflector H = I - vtvH.
subroutine zlarzt (direct, storev, n, k, v, ldv, tau, t, ldt)
ZLARZT forms the triangular factor T of a block reflector H = I - vtvH.

Detailed Description

Function Documentation

  subroutine clarzt (character direct, character storev, integer n, integer k, complex, dimension( ldv, * ) v, integer ldv, complex, dimension( * ) tau, complex, dimension( ldt, * ) t, integer ldt)

CLARZT forms the triangular factor T of a block reflector H = I - vtvH.

Purpose:

     CLARZT forms the triangular factor T of a complex block reflector
     H of order > n, which is defined as a product of k elementary
     reflectors.
     If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
     If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
     If STOREV = 'C', the vector which defines the elementary reflector
     H(i) is stored in the i-th column of the array V, and
        H  =  I - V * T * V**H
     If STOREV = 'R', the vector which defines the elementary reflector
     H(i) is stored in the i-th row of the array V, and
        H  =  I - V**H * T * V
     Currently, only STOREV = 'R' and DIRECT = 'B' are supported.

Parameters
    DIRECT

              DIRECT is CHARACTER*1
              Specifies the order in which the elementary reflectors are
              multiplied to form the block reflector:
              = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
              = 'B': H = H(k) . . . H(2) H(1) (Backward)

    STOREV

              STOREV is CHARACTER*1
              Specifies how the vectors which define the elementary
              reflectors are stored (see also Further Details):
              = 'C': columnwise                        (not supported yet)
              = 'R': rowwise

    N

              N is INTEGER
              The order of the block reflector H. N >= 0.

    K

              K is INTEGER
              The order of the triangular factor T (= the number of
              elementary reflectors). K >= 1.

    V

              V is COMPLEX array, dimension
                                   (LDV,K) if STOREV = 'C'
                                   (LDV,N) if STOREV = 'R'
              The matrix V. See further details.

    LDV

              LDV is INTEGER
              The leading dimension of the array V.
              If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.

    TAU

              TAU is COMPLEX array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i).

    T

              T is COMPLEX array, dimension (LDT,K)
              The k by k triangular factor T of the block reflector.
              If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
              lower triangular. The rest of the array is not used.

    LDT

              LDT is INTEGER
              The leading dimension of the array T. LDT >= K.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

      The shape of the matrix V and the storage of the vectors which define
      the H(i) is best illustrated by the following example with n = 5 and
      k = 3. The elements equal to 1 are not stored; the corresponding
      array elements are modified but restored on exit. The rest of the
      array is not used.
      DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
                                                  ______V_____
             ( v1 v2 v3 )                        /            \
             ( v1 v2 v3 )                      ( v1 v1 v1 v1 v1 . . . . 1 )
         V = ( v1 v2 v3 )                      ( v2 v2 v2 v2 v2 . . . 1   )
             ( v1 v2 v3 )                      ( v3 v3 v3 v3 v3 . . 1     )
             ( v1 v2 v3 )
                .  .  .
                .  .  .
                1  .  .
                   1  .
                      1
      DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
                                                            ______V_____
                1                                          /            \
                .  1                           ( 1 . . . . v1 v1 v1 v1 v1 )
                .  .  1                        ( . 1 . . . v2 v2 v2 v2 v2 )
                .  .  .                        ( . . 1 . . v3 v3 v3 v3 v3 )
                .  .  .
             ( v1 v2 v3 )
             ( v1 v2 v3 )
         V = ( v1 v2 v3 )
             ( v1 v2 v3 )
             ( v1 v2 v3 )

Definition at line 184 of file clarzt.f.

  subroutine dlarzt (character direct, character storev, integer n, integer k, double precision, dimension( ldv, * ) v, integer ldv, double precision, dimension( * ) tau, double precision, dimension( ldt, * ) t, integer ldt)

DLARZT forms the triangular factor T of a block reflector H = I - vtvH.

Purpose:

     DLARZT forms the triangular factor T of a real block reflector
     H of order > n, which is defined as a product of k elementary
     reflectors.
     If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
     If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
     If STOREV = 'C', the vector which defines the elementary reflector
     H(i) is stored in the i-th column of the array V, and
        H  =  I - V * T * V**T
     If STOREV = 'R', the vector which defines the elementary reflector
     H(i) is stored in the i-th row of the array V, and
        H  =  I - V**T * T * V
     Currently, only STOREV = 'R' and DIRECT = 'B' are supported.

Parameters
    DIRECT

              DIRECT is CHARACTER*1
              Specifies the order in which the elementary reflectors are
              multiplied to form the block reflector:
              = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
              = 'B': H = H(k) . . . H(2) H(1) (Backward)

    STOREV

              STOREV is CHARACTER*1
              Specifies how the vectors which define the elementary
              reflectors are stored (see also Further Details):
              = 'C': columnwise                        (not supported yet)
              = 'R': rowwise

    N

              N is INTEGER
              The order of the block reflector H. N >= 0.

    K

              K is INTEGER
              The order of the triangular factor T (= the number of
              elementary reflectors). K >= 1.

    V

              V is DOUBLE PRECISION array, dimension
                                   (LDV,K) if STOREV = 'C'
                                   (LDV,N) if STOREV = 'R'
              The matrix V. See further details.

    LDV

              LDV is INTEGER
              The leading dimension of the array V.
              If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.

    TAU

              TAU is DOUBLE PRECISION array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i).

    T

              T is DOUBLE PRECISION array, dimension (LDT,K)
              The k by k triangular factor T of the block reflector.
              If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
              lower triangular. The rest of the array is not used.

    LDT

              LDT is INTEGER
              The leading dimension of the array T. LDT >= K.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

      The shape of the matrix V and the storage of the vectors which define
      the H(i) is best illustrated by the following example with n = 5 and
      k = 3. The elements equal to 1 are not stored; the corresponding
      array elements are modified but restored on exit. The rest of the
      array is not used.
      DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
                                                  ______V_____
             ( v1 v2 v3 )                        /            \
             ( v1 v2 v3 )                      ( v1 v1 v1 v1 v1 . . . . 1 )
         V = ( v1 v2 v3 )                      ( v2 v2 v2 v2 v2 . . . 1   )
             ( v1 v2 v3 )                      ( v3 v3 v3 v3 v3 . . 1     )
             ( v1 v2 v3 )
                .  .  .
                .  .  .
                1  .  .
                   1  .
                      1
      DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
                                                            ______V_____
                1                                          /            \
                .  1                           ( 1 . . . . v1 v1 v1 v1 v1 )
                .  .  1                        ( . 1 . . . v2 v2 v2 v2 v2 )
                .  .  .                        ( . . 1 . . v3 v3 v3 v3 v3 )
                .  .  .
             ( v1 v2 v3 )
             ( v1 v2 v3 )
         V = ( v1 v2 v3 )
             ( v1 v2 v3 )
             ( v1 v2 v3 )

Definition at line 184 of file dlarzt.f.

  subroutine slarzt (character direct, character storev, integer n, integer k, real, dimension( ldv, * ) v, integer ldv, real, dimension( * ) tau, real, dimension( ldt, * ) t, integer ldt)

SLARZT forms the triangular factor T of a block reflector H = I - vtvH.

Purpose:

     SLARZT forms the triangular factor T of a real block reflector
     H of order > n, which is defined as a product of k elementary
     reflectors.
     If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
     If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
     If STOREV = 'C', the vector which defines the elementary reflector
     H(i) is stored in the i-th column of the array V, and
        H  =  I - V * T * V**T
     If STOREV = 'R', the vector which defines the elementary reflector
     H(i) is stored in the i-th row of the array V, and
        H  =  I - V**T * T * V
     Currently, only STOREV = 'R' and DIRECT = 'B' are supported.

Parameters
    DIRECT

              DIRECT is CHARACTER*1
              Specifies the order in which the elementary reflectors are
              multiplied to form the block reflector:
              = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
              = 'B': H = H(k) . . . H(2) H(1) (Backward)

    STOREV

              STOREV is CHARACTER*1
              Specifies how the vectors which define the elementary
              reflectors are stored (see also Further Details):
              = 'C': columnwise                        (not supported yet)
              = 'R': rowwise

    N

              N is INTEGER
              The order of the block reflector H. N >= 0.

    K

              K is INTEGER
              The order of the triangular factor T (= the number of
              elementary reflectors). K >= 1.

    V

              V is REAL array, dimension
                                   (LDV,K) if STOREV = 'C'
                                   (LDV,N) if STOREV = 'R'
              The matrix V. See further details.

    LDV

              LDV is INTEGER
              The leading dimension of the array V.
              If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.

    TAU

              TAU is REAL array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i).

    T

              T is REAL array, dimension (LDT,K)
              The k by k triangular factor T of the block reflector.
              If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
              lower triangular. The rest of the array is not used.

    LDT

              LDT is INTEGER
              The leading dimension of the array T. LDT >= K.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

      The shape of the matrix V and the storage of the vectors which define
      the H(i) is best illustrated by the following example with n = 5 and
      k = 3. The elements equal to 1 are not stored; the corresponding
      array elements are modified but restored on exit. The rest of the
      array is not used.
      DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
                                                  ______V_____
             ( v1 v2 v3 )                        /            \
             ( v1 v2 v3 )                      ( v1 v1 v1 v1 v1 . . . . 1 )
         V = ( v1 v2 v3 )                      ( v2 v2 v2 v2 v2 . . . 1   )
             ( v1 v2 v3 )                      ( v3 v3 v3 v3 v3 . . 1     )
             ( v1 v2 v3 )
                .  .  .
                .  .  .
                1  .  .
                   1  .
                      1
      DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
                                                            ______V_____
                1                                          /            \
                .  1                           ( 1 . . . . v1 v1 v1 v1 v1 )
                .  .  1                        ( . 1 . . . v2 v2 v2 v2 v2 )
                .  .  .                        ( . . 1 . . v3 v3 v3 v3 v3 )
                .  .  .
             ( v1 v2 v3 )
             ( v1 v2 v3 )
         V = ( v1 v2 v3 )
             ( v1 v2 v3 )
             ( v1 v2 v3 )

Definition at line 184 of file slarzt.f.

  subroutine zlarzt (character direct, character storev, integer n, integer k, complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension( * ) tau, complex*16, dimension( ldt, * ) t, integer ldt)

ZLARZT forms the triangular factor T of a block reflector H = I - vtvH.

Purpose:

     ZLARZT forms the triangular factor T of a complex block reflector
     H of order > n, which is defined as a product of k elementary
     reflectors.
     If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
     If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
     If STOREV = 'C', the vector which defines the elementary reflector
     H(i) is stored in the i-th column of the array V, and
        H  =  I - V * T * V**H
     If STOREV = 'R', the vector which defines the elementary reflector
     H(i) is stored in the i-th row of the array V, and
        H  =  I - V**H * T * V
     Currently, only STOREV = 'R' and DIRECT = 'B' are supported.

Parameters
    DIRECT

              DIRECT is CHARACTER*1
              Specifies the order in which the elementary reflectors are
              multiplied to form the block reflector:
              = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
              = 'B': H = H(k) . . . H(2) H(1) (Backward)

    STOREV

              STOREV is CHARACTER*1
              Specifies how the vectors which define the elementary
              reflectors are stored (see also Further Details):
              = 'C': columnwise                        (not supported yet)
              = 'R': rowwise

    N

              N is INTEGER
              The order of the block reflector H. N >= 0.

    K

              K is INTEGER
              The order of the triangular factor T (= the number of
              elementary reflectors). K >= 1.

    V

              V is COMPLEX*16 array, dimension
                                   (LDV,K) if STOREV = 'C'
                                   (LDV,N) if STOREV = 'R'
              The matrix V. See further details.

    LDV

              LDV is INTEGER
              The leading dimension of the array V.
              If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.

    TAU

              TAU is COMPLEX*16 array, dimension (K)
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i).

    T

              T is COMPLEX*16 array, dimension (LDT,K)
              The k by k triangular factor T of the block reflector.
              If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
              lower triangular. The rest of the array is not used.

    LDT

              LDT is INTEGER
              The leading dimension of the array T. LDT >= K.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

      The shape of the matrix V and the storage of the vectors which define
      the H(i) is best illustrated by the following example with n = 5 and
      k = 3. The elements equal to 1 are not stored; the corresponding
      array elements are modified but restored on exit. The rest of the
      array is not used.
      DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
                                                  ______V_____
             ( v1 v2 v3 )                        /            \
             ( v1 v2 v3 )                      ( v1 v1 v1 v1 v1 . . . . 1 )
         V = ( v1 v2 v3 )                      ( v2 v2 v2 v2 v2 . . . 1   )
             ( v1 v2 v3 )                      ( v3 v3 v3 v3 v3 . . 1     )
             ( v1 v2 v3 )
                .  .  .
                .  .  .
                1  .  .
                   1  .
                      1
      DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
                                                            ______V_____
                1                                          /            \
                .  1                           ( 1 . . . . v1 v1 v1 v1 v1 )
                .  .  1                        ( . 1 . . . v2 v2 v2 v2 v2 )
                .  .  .                        ( . . 1 . . v3 v3 v3 v3 v3 )
                .  .  .
             ( v1 v2 v3 )
             ( v1 v2 v3 )
         V = ( v1 v2 v3 )
             ( v1 v2 v3 )
             ( v1 v2 v3 )

Definition at line 184 of file zlarzt.f.

Author

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