LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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cppequ.f
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1*> \brief \b CPPEQU
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
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13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cppequ.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE CPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )
22*
23* .. Scalar Arguments ..
24* CHARACTER UPLO
25* INTEGER INFO, N
26* REAL AMAX, SCOND
27* ..
28* .. Array Arguments ..
29* REAL S( * )
30* COMPLEX AP( * )
31* ..
32*
33*
34*> \par Purpose:
35* =============
36*>
37*> \verbatim
38*>
39*> CPPEQU computes row and column scalings intended to equilibrate a
40*> Hermitian positive definite matrix A in packed storage and reduce
41*> its condition number (with respect to the two-norm). S contains the
42*> scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix
43*> B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
44*> This choice of S puts the condition number of B within a factor N of
45*> the smallest possible condition number over all possible diagonal
46*> scalings.
47*> \endverbatim
48*
49* Arguments:
50* ==========
51*
52*> \param[in] UPLO
53*> \verbatim
54*> UPLO is CHARACTER*1
55*> = 'U': Upper triangle of A is stored;
56*> = 'L': Lower triangle of A is stored.
57*> \endverbatim
58*>
59*> \param[in] N
60*> \verbatim
61*> N is INTEGER
62*> The order of the matrix A. N >= 0.
63*> \endverbatim
64*>
65*> \param[in] AP
66*> \verbatim
67*> AP is COMPLEX array, dimension (N*(N+1)/2)
68*> The upper or lower triangle of the Hermitian matrix A, packed
69*> columnwise in a linear array. The j-th column of A is stored
70*> in the array AP as follows:
71*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
72*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
73*> \endverbatim
74*>
75*> \param[out] S
76*> \verbatim
77*> S is REAL array, dimension (N)
78*> If INFO = 0, S contains the scale factors for A.
79*> \endverbatim
80*>
81*> \param[out] SCOND
82*> \verbatim
83*> SCOND is REAL
84*> If INFO = 0, S contains the ratio of the smallest S(i) to
85*> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
86*> large nor too small, it is not worth scaling by S.
87*> \endverbatim
88*>
89*> \param[out] AMAX
90*> \verbatim
91*> AMAX is REAL
92*> Absolute value of largest matrix element. If AMAX is very
93*> close to overflow or very close to underflow, the matrix
94*> should be scaled.
95*> \endverbatim
96*>
97*> \param[out] INFO
98*> \verbatim
99*> INFO is INTEGER
100*> = 0: successful exit
101*> < 0: if INFO = -i, the i-th argument had an illegal value
102*> > 0: if INFO = i, the i-th diagonal element is nonpositive.
103*> \endverbatim
104*
105* Authors:
106* ========
107*
108*> \author Univ. of Tennessee
109*> \author Univ. of California Berkeley
110*> \author Univ. of Colorado Denver
111*> \author NAG Ltd.
112*
113*> \ingroup complexOTHERcomputational
114*
115* =====================================================================
116 SUBROUTINE cppequ( UPLO, N, AP, S, SCOND, AMAX, INFO )
117*
118* -- LAPACK computational routine --
119* -- LAPACK is a software package provided by Univ. of Tennessee, --
120* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
121*
122* .. Scalar Arguments ..
123 CHARACTER UPLO
124 INTEGER INFO, N
125 REAL AMAX, SCOND
126* ..
127* .. Array Arguments ..
128 REAL S( * )
129 COMPLEX AP( * )
130* ..
131*
132* =====================================================================
133*
134* .. Parameters ..
135 REAL ONE, ZERO
136 parameter( one = 1.0e+0, zero = 0.0e+0 )
137* ..
138* .. Local Scalars ..
139 LOGICAL UPPER
140 INTEGER I, JJ
141 REAL SMIN
142* ..
143* .. External Functions ..
144 LOGICAL LSAME
145 EXTERNAL lsame
146* ..
147* .. External Subroutines ..
148 EXTERNAL xerbla
149* ..
150* .. Intrinsic Functions ..
151 INTRINSIC max, min, real, sqrt
152* ..
153* .. Executable Statements ..
154*
155* Test the input parameters.
156*
157 info = 0
158 upper = lsame( uplo, 'U' )
159 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
160 info = -1
161 ELSE IF( n.LT.0 ) THEN
162 info = -2
163 END IF
164 IF( info.NE.0 ) THEN
165 CALL xerbla( 'CPPEQU', -info )
166 RETURN
167 END IF
168*
169* Quick return if possible
170*
171 IF( n.EQ.0 ) THEN
172 scond = one
173 amax = zero
174 RETURN
175 END IF
176*
177* Initialize SMIN and AMAX.
178*
179 s( 1 ) = real( ap( 1 ) )
180 smin = s( 1 )
181 amax = s( 1 )
182*
183 IF( upper ) THEN
184*
185* UPLO = 'U': Upper triangle of A is stored.
186* Find the minimum and maximum diagonal elements.
187*
188 jj = 1
189 DO 10 i = 2, n
190 jj = jj + i
191 s( i ) = real( ap( jj ) )
192 smin = min( smin, s( i ) )
193 amax = max( amax, s( i ) )
194 10 CONTINUE
195*
196 ELSE
197*
198* UPLO = 'L': Lower triangle of A is stored.
199* Find the minimum and maximum diagonal elements.
200*
201 jj = 1
202 DO 20 i = 2, n
203 jj = jj + n - i + 2
204 s( i ) = real( ap( jj ) )
205 smin = min( smin, s( i ) )
206 amax = max( amax, s( i ) )
207 20 CONTINUE
208 END IF
209*
210 IF( smin.LE.zero ) THEN
211*
212* Find the first non-positive diagonal element and return.
213*
214 DO 30 i = 1, n
215 IF( s( i ).LE.zero ) THEN
216 info = i
217 RETURN
218 END IF
219 30 CONTINUE
220 ELSE
221*
222* Set the scale factors to the reciprocals
223* of the diagonal elements.
224*
225 DO 40 i = 1, n
226 s( i ) = one / sqrt( s( i ) )
227 40 CONTINUE
228*
229* Compute SCOND = min(S(I)) / max(S(I))
230*
231 scond = sqrt( smin ) / sqrt( amax )
232 END IF
233 RETURN
234*
235* End of CPPEQU
236*
237 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine cppequ(UPLO, N, AP, S, SCOND, AMAX, INFO)
CPPEQU
Definition: cppequ.f:117