LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
sgehd2.f
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1 *> \brief \b SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER IHI, ILO, INFO, LDA, N
25 * ..
26 * .. Array Arguments ..
27 * REAL A( LDA, * ), TAU( * ), WORK( * )
28 * ..
29 *
30 *
31 *> \par Purpose:
32 * =============
33 *>
34 *> \verbatim
35 *>
36 *> SGEHD2 reduces a real general matrix A to upper Hessenberg form H by
37 *> an orthogonal similarity transformation: Q**T * A * Q = H .
38 *> \endverbatim
39 *
40 * Arguments:
41 * ==========
42 *
43 *> \param[in] N
44 *> \verbatim
45 *> N is INTEGER
46 *> The order of the matrix A. N >= 0.
47 *> \endverbatim
48 *>
49 *> \param[in] ILO
50 *> \verbatim
51 *> ILO is INTEGER
52 *> \endverbatim
53 *>
54 *> \param[in] IHI
55 *> \verbatim
56 *> IHI is INTEGER
57 *>
58 *> It is assumed that A is already upper triangular in rows
59 *> and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
60 *> set by a previous call to SGEBAL; otherwise they should be
61 *> set to 1 and N respectively. See Further Details.
62 *> 1 <= ILO <= IHI <= max(1,N).
63 *> \endverbatim
64 *>
65 *> \param[in,out] A
66 *> \verbatim
67 *> A is REAL array, dimension (LDA,N)
68 *> On entry, the n by n general matrix to be reduced.
69 *> On exit, the upper triangle and the first subdiagonal of A
70 *> are overwritten with the upper Hessenberg matrix H, and the
71 *> elements below the first subdiagonal, with the array TAU,
72 *> represent the orthogonal matrix Q as a product of elementary
73 *> reflectors. See Further Details.
74 *> \endverbatim
75 *>
76 *> \param[in] LDA
77 *> \verbatim
78 *> LDA is INTEGER
79 *> The leading dimension of the array A. LDA >= max(1,N).
80 *> \endverbatim
81 *>
82 *> \param[out] TAU
83 *> \verbatim
84 *> TAU is REAL array, dimension (N-1)
85 *> The scalar factors of the elementary reflectors (see Further
86 *> Details).
87 *> \endverbatim
88 *>
89 *> \param[out] WORK
90 *> \verbatim
91 *> WORK is REAL array, dimension (N)
92 *> \endverbatim
93 *>
94 *> \param[out] INFO
95 *> \verbatim
96 *> INFO is INTEGER
97 *> = 0: successful exit.
98 *> < 0: if INFO = -i, the i-th argument had an illegal value.
99 *> \endverbatim
100 *
101 * Authors:
102 * ========
103 *
104 *> \author Univ. of Tennessee
105 *> \author Univ. of California Berkeley
106 *> \author Univ. of Colorado Denver
107 *> \author NAG Ltd.
108 *
109 *> \ingroup realGEcomputational
110 *
111 *> \par Further Details:
112 * =====================
113 *>
114 *> \verbatim
115 *>
116 *> The matrix Q is represented as a product of (ihi-ilo) elementary
117 *> reflectors
118 *>
119 *> Q = H(ilo) H(ilo+1) . . . H(ihi-1).
120 *>
121 *> Each H(i) has the form
122 *>
123 *> H(i) = I - tau * v * v**T
124 *>
125 *> where tau is a real scalar, and v is a real vector with
126 *> v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
127 *> exit in A(i+2:ihi,i), and tau in TAU(i).
128 *>
129 *> The contents of A are illustrated by the following example, with
130 *> n = 7, ilo = 2 and ihi = 6:
131 *>
132 *> on entry, on exit,
133 *>
134 *> ( a a a a a a a ) ( a a h h h h a )
135 *> ( a a a a a a ) ( a h h h h a )
136 *> ( a a a a a a ) ( h h h h h h )
137 *> ( a a a a a a ) ( v2 h h h h h )
138 *> ( a a a a a a ) ( v2 v3 h h h h )
139 *> ( a a a a a a ) ( v2 v3 v4 h h h )
140 *> ( a ) ( a )
141 *>
142 *> where a denotes an element of the original matrix A, h denotes a
143 *> modified element of the upper Hessenberg matrix H, and vi denotes an
144 *> element of the vector defining H(i).
145 *> \endverbatim
146 *>
147 * =====================================================================
148  SUBROUTINE sgehd2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
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 *
222  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine sgehd2(N, ILO, IHI, A, LDA, TAU, WORK, INFO)
SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
Definition: sgehd2.f:149
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