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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