 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ zpoequb()

 subroutine zpoequb ( integer N, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, integer INFO )

ZPOEQUB

Purpose:
``` ZPOEQUB computes row and column scalings intended to equilibrate a
Hermitian positive definite matrix A and reduce its condition number
(with respect to the two-norm).  S contains the scale factors,
S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
choice of S puts the condition number of B within a factor N of the
smallest possible condition number over all possible diagonal
scalings.

This routine differs from ZPOEQU by restricting the scaling factors
to a power of the radix.  Barring over- and underflow, scaling by
these factors introduces no additional rounding errors.  However, the
scaled diagonal entries are no longer approximately 1 but lie
Parameters
 [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in] A ``` A is COMPLEX*16 array, dimension (LDA,N) The N-by-N Hermitian positive definite matrix whose scaling factors are to be computed. Only the diagonal elements of A are referenced.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [out] S ``` S is DOUBLE PRECISION array, dimension (N) If INFO = 0, S contains the scale factors for A.``` [out] SCOND ``` SCOND is DOUBLE PRECISION If INFO = 0, S contains the ratio of the smallest S(i) to the largest S(i). If SCOND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by S.``` [out] AMAX ``` AMAX is DOUBLE PRECISION Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the i-th diagonal element is nonpositive.```

Definition at line 118 of file zpoequb.f.

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*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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