 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ 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)
XERBLA
Definition: xerbla.f:60
subroutine slarfg(N, ALPHA, X, INCX, TAU)
SLARFG generates an elementary reflector (Householder matrix).
Definition: slarfg.f:106
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
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