LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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zpoequb.f
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1*> \brief \b ZPOEQUB
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download ZPOEQUB + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zpoequb.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zpoequb.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zpoequb.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE ZPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )
22*
23* .. Scalar Arguments ..
24* INTEGER INFO, LDA, N
25* DOUBLE PRECISION AMAX, SCOND
26* ..
27* .. Array Arguments ..
28* COMPLEX*16 A( LDA, * )
29* DOUBLE PRECISION S( * )
30* ..
31*
32*
33*> \par Purpose:
34* =============
35*>
36*> \verbatim
37*>
38*> ZPOEQUB computes row and column scalings intended to equilibrate a
39*> Hermitian positive definite matrix A and reduce its condition number
40*> (with respect to the two-norm). S contains the scale factors,
41*> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
42*> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
43*> choice of S puts the condition number of B within a factor N of the
44*> smallest possible condition number over all possible diagonal
45*> scalings.
46*>
47*> This routine differs from ZPOEQU by restricting the scaling factors
48*> to a power of the radix. Barring over- and underflow, scaling by
49*> these factors introduces no additional rounding errors. However, the
50*> scaled diagonal entries are no longer approximately 1 but lie
51*> between sqrt(radix) and 1/sqrt(radix).
52*> \endverbatim
53*
54* Arguments:
55* ==========
56*
57*> \param[in] N
58*> \verbatim
59*> N is INTEGER
60*> The order of the matrix A. N >= 0.
61*> \endverbatim
62*>
63*> \param[in] A
64*> \verbatim
65*> A is COMPLEX*16 array, dimension (LDA,N)
66*> The N-by-N Hermitian positive definite matrix whose scaling
67*> factors are to be computed. Only the diagonal elements of A
68*> are referenced.
69*> \endverbatim
70*>
71*> \param[in] LDA
72*> \verbatim
73*> LDA is INTEGER
74*> The leading dimension of the array A. LDA >= max(1,N).
75*> \endverbatim
76*>
77*> \param[out] S
78*> \verbatim
79*> S is DOUBLE PRECISION array, dimension (N)
80*> If INFO = 0, S contains the scale factors for A.
81*> \endverbatim
82*>
83*> \param[out] SCOND
84*> \verbatim
85*> SCOND is DOUBLE PRECISION
86*> If INFO = 0, S contains the ratio of the smallest S(i) to
87*> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
88*> large nor too small, it is not worth scaling by S.
89*> \endverbatim
90*>
91*> \param[out] AMAX
92*> \verbatim
93*> AMAX is DOUBLE PRECISION
94*> Absolute value of largest matrix element. If AMAX is very
95*> close to overflow or very close to underflow, the matrix
96*> should be scaled.
97*> \endverbatim
98*>
99*> \param[out] INFO
100*> \verbatim
101*> INFO is INTEGER
102*> = 0: successful exit
103*> < 0: if INFO = -i, the i-th argument had an illegal value
104*> > 0: if INFO = i, the i-th diagonal element is nonpositive.
105*> \endverbatim
106*
107* Authors:
108* ========
109*
110*> \author Univ. of Tennessee
111*> \author Univ. of California Berkeley
112*> \author Univ. of Colorado Denver
113*> \author NAG Ltd.
114*
115*> \ingroup complex16POcomputational
116*
117* =====================================================================
118 SUBROUTINE zpoequb( N, A, LDA, S, SCOND, AMAX, INFO )
119*
120* -- LAPACK computational routine --
121* -- LAPACK is a software package provided by Univ. of Tennessee, --
122* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
123*
124* .. Scalar Arguments ..
125 INTEGER INFO, LDA, N
126 DOUBLE PRECISION AMAX, SCOND
127* ..
128* .. Array Arguments ..
129 COMPLEX*16 A( LDA, * )
130 DOUBLE PRECISION S( * )
131* ..
132*
133* =====================================================================
134*
135* .. Parameters ..
136 DOUBLE PRECISION ZERO, ONE
137 parameter( zero = 0.0d+0, one = 1.0d+0 )
138* ..
139* .. Local Scalars ..
140 INTEGER I
141 DOUBLE PRECISION SMIN, BASE, TMP
142* ..
143* .. External Functions ..
144 DOUBLE PRECISION DLAMCH
145 EXTERNAL dlamch
146* ..
147* .. External Subroutines ..
148 EXTERNAL xerbla
149* ..
150* .. Intrinsic Functions ..
151 INTRINSIC max, min, sqrt, log, int, real, dimag
152* ..
153* .. Executable Statements ..
154*
155* Test the input parameters.
156*
157* Positive definite only performs 1 pass of equilibration.
158*
159 info = 0
160 IF( n.LT.0 ) THEN
161 info = -1
162 ELSE IF( lda.LT.max( 1, n ) ) THEN
163 info = -3
164 END IF
165 IF( info.NE.0 ) THEN
166 CALL xerbla( 'ZPOEQUB', -info )
167 RETURN
168 END IF
169*
170* Quick return if possible.
171*
172 IF( n.EQ.0 ) THEN
173 scond = one
174 amax = zero
175 RETURN
176 END IF
177
178 base = dlamch( 'B' )
179 tmp = -0.5d+0 / log( base )
180*
181* Find the minimum and maximum diagonal elements.
182*
183 s( 1 ) = dble( a( 1, 1 ) )
184 smin = s( 1 )
185 amax = s( 1 )
186 DO 10 i = 2, n
187 s( i ) = dble( a( i, i ) )
188 smin = min( smin, s( i ) )
189 amax = max( amax, s( i ) )
190 10 CONTINUE
191*
192 IF( smin.LE.zero ) THEN
193*
194* Find the first non-positive diagonal element and return.
195*
196 DO 20 i = 1, n
197 IF( s( i ).LE.zero ) THEN
198 info = i
199 RETURN
200 END IF
201 20 CONTINUE
202 ELSE
203*
204* Set the scale factors to the reciprocals
205* of the diagonal elements.
206*
207 DO 30 i = 1, n
208 s( i ) = base ** int( tmp * log( s( i ) ) )
209 30 CONTINUE
210*
211* Compute SCOND = min(S(I)) / max(S(I)).
212*
213 scond = sqrt( smin ) / sqrt( amax )
214 END IF
215*
216 RETURN
217*
218* End of ZPOEQUB
219*
220 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zpoequb(N, A, LDA, S, SCOND, AMAX, INFO)
ZPOEQUB
Definition: zpoequb.f:119