.TH "larzt" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
larzt \- larzt: generate T matrix
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBclarzt\fP (direct, storev, n, k, v, ldv, tau, t, ldt)"
.br
.RI "\fBCLARZT\fP forms the triangular factor T of a block reflector H = I - vtvH\&. "
.ti -1c
.RI "subroutine \fBdlarzt\fP (direct, storev, n, k, v, ldv, tau, t, ldt)"
.br
.RI "\fBDLARZT\fP forms the triangular factor T of a block reflector H = I - vtvH\&. "
.ti -1c
.RI "subroutine \fBslarzt\fP (direct, storev, n, k, v, ldv, tau, t, ldt)"
.br
.RI "\fBSLARZT\fP forms the triangular factor T of a block reflector H = I - vtvH\&. "
.ti -1c
.RI "subroutine \fBzlarzt\fP (direct, storev, n, k, v, ldv, tau, t, ldt)"
.br
.RI "\fBZLARZT\fP forms the triangular factor T of a block reflector H = I - vtvH\&. "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "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)"

.PP
\fBCLARZT\fP forms the triangular factor T of a block reflector H = I - vtvH\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 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\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIDIRECT\fP 
.PP
.nf
          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)
.fi
.PP
.br
\fISTOREV\fP 
.PP
.nf
          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
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the block reflector H\&. N >= 0\&.
.fi
.PP
.br
\fIK\fP 
.PP
.nf
          K is INTEGER
          The order of the triangular factor T (= the number of
          elementary reflectors)\&. K >= 1\&.
.fi
.PP
.br
\fIV\fP 
.PP
.nf
          V is COMPLEX array, dimension
                               (LDV,K) if STOREV = 'C'
                               (LDV,N) if STOREV = 'R'
          The matrix V\&. See further details\&.
.fi
.PP
.br
\fILDV\fP 
.PP
.nf
          LDV is INTEGER
          The leading dimension of the array V\&.
          If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K\&.
.fi
.PP
.br
\fITAU\fP 
.PP
.nf
          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i)\&.
.fi
.PP
.br
\fIT\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDT\fP 
.PP
.nf
          LDT is INTEGER
          The leading dimension of the array T\&. LDT >= K\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBContributors:\fP
.RS 4
A\&. Petitet, Computer Science Dept\&., Univ\&. of Tenn\&., Knoxville, USA 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
  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 )
.fi
.PP
 
.RE
.PP

.SS "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)"

.PP
\fBDLARZT\fP forms the triangular factor T of a block reflector H = I - vtvH\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 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\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIDIRECT\fP 
.PP
.nf
          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)
.fi
.PP
.br
\fISTOREV\fP 
.PP
.nf
          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
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the block reflector H\&. N >= 0\&.
.fi
.PP
.br
\fIK\fP 
.PP
.nf
          K is INTEGER
          The order of the triangular factor T (= the number of
          elementary reflectors)\&. K >= 1\&.
.fi
.PP
.br
\fIV\fP 
.PP
.nf
          V is DOUBLE PRECISION array, dimension
                               (LDV,K) if STOREV = 'C'
                               (LDV,N) if STOREV = 'R'
          The matrix V\&. See further details\&.
.fi
.PP
.br
\fILDV\fP 
.PP
.nf
          LDV is INTEGER
          The leading dimension of the array V\&.
          If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K\&.
.fi
.PP
.br
\fITAU\fP 
.PP
.nf
          TAU is DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i)\&.
.fi
.PP
.br
\fIT\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDT\fP 
.PP
.nf
          LDT is INTEGER
          The leading dimension of the array T\&. LDT >= K\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBContributors:\fP
.RS 4
A\&. Petitet, Computer Science Dept\&., Univ\&. of Tenn\&., Knoxville, USA 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
  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 )
.fi
.PP
 
.RE
.PP

.SS "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)"

.PP
\fBSLARZT\fP forms the triangular factor T of a block reflector H = I - vtvH\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 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\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIDIRECT\fP 
.PP
.nf
          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)
.fi
.PP
.br
\fISTOREV\fP 
.PP
.nf
          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
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the block reflector H\&. N >= 0\&.
.fi
.PP
.br
\fIK\fP 
.PP
.nf
          K is INTEGER
          The order of the triangular factor T (= the number of
          elementary reflectors)\&. K >= 1\&.
.fi
.PP
.br
\fIV\fP 
.PP
.nf
          V is REAL array, dimension
                               (LDV,K) if STOREV = 'C'
                               (LDV,N) if STOREV = 'R'
          The matrix V\&. See further details\&.
.fi
.PP
.br
\fILDV\fP 
.PP
.nf
          LDV is INTEGER
          The leading dimension of the array V\&.
          If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K\&.
.fi
.PP
.br
\fITAU\fP 
.PP
.nf
          TAU is REAL array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i)\&.
.fi
.PP
.br
\fIT\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDT\fP 
.PP
.nf
          LDT is INTEGER
          The leading dimension of the array T\&. LDT >= K\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBContributors:\fP
.RS 4
A\&. Petitet, Computer Science Dept\&., Univ\&. of Tenn\&., Knoxville, USA 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
  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 )
.fi
.PP
 
.RE
.PP

.SS "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)"

.PP
\fBZLARZT\fP forms the triangular factor T of a block reflector H = I - vtvH\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 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\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIDIRECT\fP 
.PP
.nf
          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)
.fi
.PP
.br
\fISTOREV\fP 
.PP
.nf
          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
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the block reflector H\&. N >= 0\&.
.fi
.PP
.br
\fIK\fP 
.PP
.nf
          K is INTEGER
          The order of the triangular factor T (= the number of
          elementary reflectors)\&. K >= 1\&.
.fi
.PP
.br
\fIV\fP 
.PP
.nf
          V is COMPLEX*16 array, dimension
                               (LDV,K) if STOREV = 'C'
                               (LDV,N) if STOREV = 'R'
          The matrix V\&. See further details\&.
.fi
.PP
.br
\fILDV\fP 
.PP
.nf
          LDV is INTEGER
          The leading dimension of the array V\&.
          If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K\&.
.fi
.PP
.br
\fITAU\fP 
.PP
.nf
          TAU is COMPLEX*16 array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i)\&.
.fi
.PP
.br
\fIT\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fILDT\fP 
.PP
.nf
          LDT is INTEGER
          The leading dimension of the array T\&. LDT >= K\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBContributors:\fP
.RS 4
A\&. Petitet, Computer Science Dept\&., Univ\&. of Tenn\&., Knoxville, USA 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
  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 )
.fi
.PP
 
.RE
.PP

.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.