larft(3)

NAME
larft - larft: generate T matrix

SYNOPSIS

  Functions

recursive subroutine clarft (direct, storev, n, k, v, ldv, tau, t, ldt)
CLARFT forms the triangular factor T of a block reflector H = I - vtvH
recursive subroutine dlarft (direct, storev, n, k, v, ldv, tau, t, ldt)
DLARFT forms the triangular factor T of a block reflector H = I - vtvH
recursive subroutine slarft (direct, storev, n, k, v, ldv, tau, t, ldt)
SLARFT forms the triangular factor T of a block reflector H = I - vtvH
recursive subroutine zlarft (direct, storev, n, k, v, ldv, tau, t, ldt)
ZLARFT forms the triangular factor T of a block reflector H = I - vtvH

Detailed Description

Function Documentation

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

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

Purpose:

    !>
    !> CLARFT 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
    !>

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)
    !>          = '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
    !>          = '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.

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.
    !>
    !>  DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
    !>
    !>               V = (  1       )                 V = (  1 v1 v1 v1 v1 )
    !>                   ( v1  1    )                     (     1 v2 v2 v2 )
    !>                   ( v1 v2  1 )                     (        1 v3 v3 )
    !>                   ( v1 v2 v3 )
    !>                   ( v1 v2 v3 )
    !>
    !>  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
    !>
    !>               V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
    !>                   ( v1 v2 v3 )                     ( v2 v2 v2  1    )
    !>                   (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
    !>                   (     1 v3 )
    !>                   (        1 )
    !>

  recursive subroutine dlarft (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)

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

Purpose:

    !>
    !> DLARFT 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
    !>

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)
    !>          = '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
    !>          = '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.

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.
    !>
    !>  DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
    !>
    !>               V = (  1       )                 V = (  1 v1 v1 v1 v1 )
    !>                   ( v1  1    )                     (     1 v2 v2 v2 )
    !>                   ( v1 v2  1 )                     (        1 v3 v3 )
    !>                   ( v1 v2 v3 )
    !>                   ( v1 v2 v3 )
    !>
    !>  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
    !>
    !>               V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
    !>                   ( v1 v2 v3 )                     ( v2 v2 v2  1    )
    !>                   (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
    !>                   (     1 v3 )
    !>                   (        1 )
    !>

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

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

Purpose:

    !>
    !> SLARFT 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
    !>

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)
    !>          = '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
    !>          = '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

    Johnathan Rhyne, Univ. of Colorado Denver (original author, 2024)

    NAG Ltd.

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.
    !>
    !>  DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
    !>
    !>               V = (  1       )                 V = (  1 v1 v1 v1 v1 )
    !>                   ( v1  1    )                     (     1 v2 v2 v2 )
    !>                   ( v1 v2  1 )                     (        1 v3 v3 )
    !>                   ( v1 v2 v3 )
    !>                   ( v1 v2 v3 )
    !>
    !>  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
    !>
    !>               V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
    !>                   ( v1 v2 v3 )                     ( v2 v2 v2  1    )
    !>                   (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
    !>                   (     1 v3 )
    !>                   (        1 )
    !>

  recursive subroutine zlarft (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)

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

Purpose:

    !>
    !> ZLARFT 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
    !>

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)
    !>          = '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
    !>          = '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.

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.
    !>
    !>  DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
    !>
    !>               V = (  1       )                 V = (  1 v1 v1 v1 v1 )
    !>                   ( v1  1    )                     (     1 v2 v2 v2 )
    !>                   ( v1 v2  1 )                     (        1 v3 v3 )
    !>                   ( v1 v2 v3 )
    !>                   ( v1 v2 v3 )
    !>
    !>  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
    !>
    !>               V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
    !>                   ( v1 v2 v3 )                     ( v2 v2 v2  1    )
    !>                   (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
    !>                   (     1 v3 )
    !>                   (        1 )
    !>

Author

Generated automatically by Doxygen for LAPACK from the source code.