LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine dgehd2 ( integer N, integer ILO, integer IHI, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer INFO )

DGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

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Purpose:
``` DGEHD2 reduces a real general matrix A to upper Hessenberg form H by
an orthogonal similarity transformation:  Q**T * A * Q = H .```
Parameters
 [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in] ILO ` ILO is INTEGER` [in] IHI ``` IHI is INTEGER It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to DGEBAL; otherwise they should be set to 1 and N respectively. See Further Details. 1 <= ILO <= IHI <= max(1,N).``` [in,out] A ``` A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the n by n general matrix to be reduced. On exit, the upper triangle and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H, and the elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [out] TAU ``` TAU is DOUBLE PRECISION array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).``` [out] WORK ` WORK is DOUBLE PRECISION array, dimension (N)` [out] INFO ``` INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.```
Date
September 2012
Further Details:
```  The matrix Q is represented as a product of (ihi-ilo) elementary
reflectors

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Each H(i) has the form

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

where tau is a real scalar, and v is a real vector with
v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
exit in A(i+2:ihi,i), and tau in TAU(i).

The contents of A are illustrated by the following example, with
n = 7, ilo = 2 and ihi = 6:

on entry,                        on exit,

( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      h   h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
(     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
(                         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).```

Definition at line 151 of file dgehd2.f.

151 *
152 * -- LAPACK computational routine (version 3.4.2) --
153 * -- LAPACK is a software package provided by Univ. of Tennessee, --
154 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
155 * September 2012
156 *
157 * .. Scalar Arguments ..
158  INTEGER ihi, ilo, info, lda, n
159 * ..
160 * .. Array Arguments ..
161  DOUBLE PRECISION a( lda, * ), tau( * ), work( * )
162 * ..
163 *
164 * =====================================================================
165 *
166 * .. Parameters ..
167  DOUBLE PRECISION one
168  parameter ( one = 1.0d+0 )
169 * ..
170 * .. Local Scalars ..
171  INTEGER i
172  DOUBLE PRECISION aii
173 * ..
174 * .. External Subroutines ..
175  EXTERNAL dlarf, dlarfg, xerbla
176 * ..
177 * .. Intrinsic Functions ..
178  INTRINSIC max, min
179 * ..
180 * .. Executable Statements ..
181 *
182 * Test the input parameters
183 *
184  info = 0
185  IF( n.LT.0 ) THEN
186  info = -1
187  ELSE IF( ilo.LT.1 .OR. ilo.GT.max( 1, n ) ) THEN
188  info = -2
189  ELSE IF( ihi.LT.min( ilo, n ) .OR. ihi.GT.n ) THEN
190  info = -3
191  ELSE IF( lda.LT.max( 1, n ) ) THEN
192  info = -5
193  END IF
194  IF( info.NE.0 ) THEN
195  CALL xerbla( 'DGEHD2', -info )
196  RETURN
197  END IF
198 *
199  DO 10 i = ilo, ihi - 1
200 *
201 * Compute elementary reflector H(i) to annihilate A(i+2:ihi,i)
202 *
203  CALL dlarfg( ihi-i, a( i+1, i ), a( min( i+2, n ), i ), 1,
204  \$ tau( i ) )
205  aii = a( i+1, i )
206  a( i+1, i ) = one
207 *
208 * Apply H(i) to A(1:ihi,i+1:ihi) from the right
209 *
210  CALL dlarf( 'Right', ihi, ihi-i, a( i+1, i ), 1, tau( i ),
211  \$ a( 1, i+1 ), lda, work )
212 *
213 * Apply H(i) to A(i+1:ihi,i+1:n) from the left
214 *
215  CALL dlarf( 'Left', ihi-i, n-i, a( i+1, i ), 1, tau( i ),
216  \$ a( i+1, i+1 ), lda, work )
217 *
218  a( i+1, i ) = aii
219  10 CONTINUE
220 *
221  RETURN
222 *
223 * End of DGEHD2
224 *
subroutine dlarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
DLARF applies an elementary reflector to a general rectangular matrix.
Definition: dlarf.f:126
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine dlarfg(N, ALPHA, X, INCX, TAU)
DLARFG generates an elementary reflector (Householder matrix).
Definition: dlarfg.f:108

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