larft(3)

NAME
larft - larft: generate T matrix

SYNOPSIS

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

  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 )

Definition at line 162 of file clarft.f.

  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 )

Definition at line 162 of file dlarft.f.

  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

    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 )

Definition at line 162 of file slarft.f.

  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 )

Definition at line 162 of file zlarft.f.

Author

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