.TH "clahr2.f" 3 "Tue Nov 14 2017" "Version 3.8.0" "LAPACK" \" -*- nroff -*-
.ad l
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.SH NAME
clahr2.f
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBclahr2\fP (\fBN\fP, K, NB, A, \fBLDA\fP, TAU, T, LDT, Y, LDY)"
.br
.RI "\fBCLAHR2\fP reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A\&. "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "subroutine clahr2 (integer N, integer K, integer NB, complex, dimension( lda, * ) A, integer LDA, complex, dimension( nb ) TAU, complex, dimension( ldt, nb ) T, integer LDT, complex, dimension( ldy, nb ) Y, integer LDY)"

.PP
\fBCLAHR2\fP reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A\&.  
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\fBPurpose: \fP
.RS 4

.PP
.nf
 CLAHR2 reduces the first NB columns of A complex general n-BY-(n-k+1)
 matrix A so that elements below the k-th subdiagonal are zero. The
 reduction is performed by an unitary similarity transformation
 Q**H * A * Q. The routine returns the matrices V and T which determine
 Q as a block reflector I - V*T*v**H, and also the matrix Y = A * V * T.

 This is an auxiliary routine called by CGEHRD.
.fi
.PP
 
.RE
.PP
\fBParameters:\fP
.RS 4
\fIN\fP 
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.nf
          N is INTEGER
          The order of the matrix A.
.fi
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.br
\fIK\fP 
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          K is INTEGER
          The offset for the reduction. Elements below the k-th
          subdiagonal in the first NB columns are reduced to zero.
          K < N.
.fi
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\fINB\fP 
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.nf
          NB is INTEGER
          The number of columns to be reduced.
.fi
.PP
.br
\fIA\fP 
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          A is COMPLEX array, dimension (LDA,N-K+1)
          On entry, the n-by-(n-k+1) general matrix A.
          On exit, the elements on and above the k-th subdiagonal in
          the first NB columns are overwritten with the corresponding
          elements of the reduced matrix; the elements below the k-th
          subdiagonal, with the array TAU, represent the matrix Q as a
          product of elementary reflectors. The other columns of A are
          unchanged. See Further Details.
.fi
.PP
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\fILDA\fP 
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          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).
.fi
.PP
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\fITAU\fP 
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          TAU is COMPLEX array, dimension (NB)
          The scalar factors of the elementary reflectors. See Further
          Details.
.fi
.PP
.br
\fIT\fP 
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.nf
          T is COMPLEX array, dimension (LDT,NB)
          The upper triangular matrix T.
.fi
.PP
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\fILDT\fP 
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.nf
          LDT is INTEGER
          The leading dimension of the array T.  LDT >= NB.
.fi
.PP
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\fIY\fP 
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          Y is COMPLEX array, dimension (LDY,NB)
          The n-by-nb matrix Y.
.fi
.PP
.br
\fILDY\fP 
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.nf
          LDY is INTEGER
          The leading dimension of the array Y. LDY >= N.
.fi
.PP
 
.RE
.PP
\fBAuthor:\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
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Univ\&. of Colorado Denver 
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NAG Ltd\&. 
.RE
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\fBDate:\fP
.RS 4
December 2016 
.RE
.PP
\fBFurther Details: \fP
.RS 4

.PP
.nf
  The matrix Q is represented as a product of nb elementary reflectors

     Q = H(1) H(2) . . . H(nb).

  Each H(i) has the form

     H(i) = I - tau * v * v**H

  where tau is a complex scalar, and v is a complex vector with
  v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
  A(i+k+1:n,i), and tau in TAU(i).

  The elements of the vectors v together form the (n-k+1)-by-nb matrix
  V which is needed, with T and Y, to apply the transformation to the
  unreduced part of the matrix, using an update of the form:
  A := (I - V*T*V**H) * (A - Y*V**H).

  The contents of A on exit are illustrated by the following example
  with n = 7, k = 3 and nb = 2:

     ( a   a   a   a   a )
     ( a   a   a   a   a )
     ( a   a   a   a   a )
     ( h   h   a   a   a )
     ( v1  h   a   a   a )
     ( v1  v2  a   a   a )
     ( v1  v2  a   a   a )

  where a denotes an element of the original matrix A, h denotes a
  modified element of the upper Hessenberg matrix H, and vi denotes an
  element of the vector defining H(i).

  This subroutine is a slight modification of LAPACK-3.0's DLAHRD
  incorporating improvements proposed by Quintana-Orti and Van de
  Gejin. Note that the entries of A(1:K,2:NB) differ from those
  returned by the original LAPACK-3.0's DLAHRD routine. (This
  subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)
.fi
.PP
 
.RE
.PP
\fBReferences: \fP
.RS 4
Gregorio Quintana-Orti and Robert van de Geijn, 'Improving the
  performance of reduction to Hessenberg form,' ACM Transactions on Mathematical Software, 32(2):180-194, June 2006\&. 
.RE
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Definition at line 183 of file clahr2\&.f\&.
.SH "Author"
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Generated automatically by Doxygen for LAPACK from the source code\&.