LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ sgehd2()

 subroutine sgehd2 ( integer n, integer ilo, integer ihi, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work, integer info )

SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

Purpose:
``` SGEHD2 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 SGEBAL; otherwise they should be set to 1 and N respectively. See Further Details. 1 <= ILO <= IHI <= max(1,N).``` [in,out] A ``` A is REAL 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 REAL array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).``` [out] WORK ` WORK is REAL array, dimension (N)` [out] INFO ``` INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.```
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 sgehd2.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 REAL A( LDA, * ), TAU( * ), WORK( * )
159* ..
160*
161* =====================================================================
162*
163* .. Parameters ..
164 REAL ONE
165 parameter( one = 1.0e+0 )
166* ..
167* .. Local Scalars ..
168 INTEGER I
169 REAL AII
170* ..
171* .. External Subroutines ..
172 EXTERNAL slarf, slarfg, 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( 'SGEHD2', -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 slarfg( 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 slarf( '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 slarf( '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 SGEHD2
221*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine slarf(side, m, n, v, incv, tau, c, ldc, work)
SLARF applies an elementary reflector to a general rectangular matrix.
Definition slarf.f:124
subroutine slarfg(n, alpha, x, incx, tau)
SLARFG generates an elementary reflector (Householder matrix).
Definition slarfg.f:106
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