LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ dgehd2()

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.

Download DGEHD2 + dependencies [TGZ] [ZIP] [TXT]

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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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 148 of file dgehd2.f.

149*
150* -- LAPACK computational routine --
151* -- LAPACK is a software package provided by Univ. of Tennessee, --
152* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
153*
154* .. Scalar Arguments ..
155 INTEGER IHI, ILO, INFO, LDA, N
156* ..
157* .. Array Arguments ..
158 DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
159* ..
160*
161* =====================================================================
162*
163* .. Parameters ..
164 DOUBLE PRECISION ONE
165 parameter( one = 1.0d+0 )
166* ..
167* .. Local Scalars ..
168 INTEGER I
169 DOUBLE PRECISION AII
170* ..
171* .. External Subroutines ..
172 EXTERNAL dlarf, dlarfg, xerbla
173* ..
174* .. Intrinsic Functions ..
175 INTRINSIC max, min
176* ..
177* .. Executable Statements ..
178*
179* Test the input parameters
180*
181 info = 0
182 IF( n.LT.0 ) THEN
183 info = -1
184 ELSE IF( ilo.LT.1 .OR. ilo.GT.max( 1, n ) ) THEN
185 info = -2
186 ELSE IF( ihi.LT.min( ilo, n ) .OR. ihi.GT.n ) THEN
187 info = -3
188 ELSE IF( lda.LT.max( 1, n ) ) THEN
189 info = -5
190 END IF
191 IF( info.NE.0 ) THEN
192 CALL xerbla( 'DGEHD2', -info )
193 RETURN
194 END IF
195*
196 DO 10 i = ilo, ihi - 1
197*
198* Compute elementary reflector H(i) to annihilate A(i+2:ihi,i)
199*
200 CALL dlarfg( ihi-i, a( i+1, i ), a( min( i+2, n ), i ), 1,
201 $ tau( i ) )
202 aii = a( i+1, i )
203 a( i+1, i ) = one
204*
205* Apply H(i) to A(1:ihi,i+1:ihi) from the right
206*
207 CALL dlarf( 'Right', ihi, ihi-i, a( i+1, i ), 1, tau( i ),
208 $ a( 1, i+1 ), lda, work )
209*
210* Apply H(i) to A(i+1:ihi,i+1:n) from the left
211*
212 CALL dlarf( 'Left', ihi-i, n-i, a( i+1, i ), 1, tau( i ),
213 $ a( i+1, i+1 ), lda, work )
214*
215 a( i+1, i ) = aii
216 10 CONTINUE
217*
218 RETURN
219*
220* End of DGEHD2
221*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dlarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
DLARF applies an elementary reflector to a general rectangular matrix.
Definition: dlarf.f:124
subroutine dlarfg(N, ALPHA, X, INCX, TAU)
DLARFG generates an elementary reflector (Householder matrix).
Definition: dlarfg.f:106
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