LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ dpoequb()

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

DPOEQUB

Download DPOEQUB + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 DPOEQUB computes row and column scalings intended to equilibrate a
 symmetric 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 DPOEQU 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
 between sqrt(radix) and 1/sqrt(radix).
Parameters
[in]N
          N is INTEGER
          The order of the matrix A.  N >= 0.
[in]A
          A is DOUBLE PRECISION array, dimension (LDA,N)
          The N-by-N symmetric 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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 117 of file dpoequb.f.

118*
119* -- LAPACK computational routine --
120* -- LAPACK is a software package provided by Univ. of Tennessee, --
121* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
122*
123* .. Scalar Arguments ..
124 INTEGER INFO, LDA, N
125 DOUBLE PRECISION AMAX, SCOND
126* ..
127* .. Array Arguments ..
128 DOUBLE PRECISION A( LDA, * ), S( * )
129* ..
130*
131* =====================================================================
132*
133* .. Parameters ..
134 DOUBLE PRECISION ZERO, ONE
135 parameter( zero = 0.0d+0, one = 1.0d+0 )
136* ..
137* .. Local Scalars ..
138 INTEGER I
139 DOUBLE PRECISION SMIN, BASE, TMP
140* ..
141* .. External Functions ..
142 DOUBLE PRECISION DLAMCH
143 EXTERNAL dlamch
144* ..
145* .. External Subroutines ..
146 EXTERNAL xerbla
147* ..
148* .. Intrinsic Functions ..
149 INTRINSIC max, min, sqrt, log, int
150* ..
151* .. Executable Statements ..
152*
153* Test the input parameters.
154*
155* Positive definite only performs 1 pass of equilibration.
156*
157 info = 0
158 IF( n.LT.0 ) THEN
159 info = -1
160 ELSE IF( lda.LT.max( 1, n ) ) THEN
161 info = -3
162 END IF
163 IF( info.NE.0 ) THEN
164 CALL xerbla( 'DPOEQUB', -info )
165 RETURN
166 END IF
167*
168* Quick return if possible.
169*
170 IF( n.EQ.0 ) THEN
171 scond = one
172 amax = zero
173 RETURN
174 END IF
175
176 base = dlamch( 'B' )
177 tmp = -0.5d+0 / log( base )
178*
179* Find the minimum and maximum diagonal elements.
180*
181 s( 1 ) = a( 1, 1 )
182 smin = s( 1 )
183 amax = s( 1 )
184 DO 10 i = 2, n
185 s( i ) = a( i, i )
186 smin = min( smin, s( i ) )
187 amax = max( amax, s( i ) )
188 10 CONTINUE
189*
190 IF( smin.LE.zero ) THEN
191*
192* Find the first non-positive diagonal element and return.
193*
194 DO 20 i = 1, n
195 IF( s( i ).LE.zero ) THEN
196 info = i
197 RETURN
198 END IF
199 20 CONTINUE
200 ELSE
201*
202* Set the scale factors to the reciprocals
203* of the diagonal elements.
204*
205 DO 30 i = 1, n
206 s( i ) = base ** int( tmp * log( s( i ) ) )
207 30 CONTINUE
208*
209* Compute SCOND = min(S(I)) / max(S(I)).
210*
211 scond = sqrt( smin ) / sqrt( amax )
212 END IF
213*
214 RETURN
215*
216* End of DPOEQUB
217*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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