LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ spoequb()

 subroutine spoequb ( integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) S, real SCOND, real AMAX, integer INFO )

SPOEQUB

Purpose:
``` SPOEQUB 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 SPOEQU 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 REAL 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 REAL array, dimension (N) If INFO = 0, S contains the scale factors for A.``` [out] SCOND ``` SCOND is REAL 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 REAL 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.```
Date
December 2016

Definition at line 120 of file spoequb.f.

120 *
121 * -- LAPACK computational routine (version 3.7.0) --
122 * -- LAPACK is a software package provided by Univ. of Tennessee, --
123 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
124 * December 2016
125 *
126 * .. Scalar Arguments ..
127  INTEGER info, lda, n
128  REAL amax, scond
129 * ..
130 * .. Array Arguments ..
131  REAL a( lda, * ), s( * )
132 * ..
133 *
134 * =====================================================================
135 *
136 * .. Parameters ..
137  REAL zero, one
138  parameter( zero = 0.0e+0, one = 1.0e+0 )
139 * ..
140 * .. Local Scalars ..
141  INTEGER i
142  REAL smin, base, tmp
143 * ..
144 * .. External Functions ..
145  REAL slamch
146  EXTERNAL slamch
147 * ..
148 * .. External Subroutines ..
149  EXTERNAL xerbla
150 * ..
151 * .. Intrinsic Functions ..
152  INTRINSIC max, min, sqrt, log, int
153 * ..
154 * .. Executable Statements ..
155 *
156 * Test the input parameters.
157 *
158 * Positive definite only performs 1 pass of equilibration.
159 *
160  info = 0
161  IF( n.LT.0 ) THEN
162  info = -1
163  ELSE IF( lda.LT.max( 1, n ) ) THEN
164  info = -3
165  END IF
166  IF( info.NE.0 ) THEN
167  CALL xerbla( 'SPOEQUB', -info )
168  RETURN
169  END IF
170 *
171 * Quick return if possible.
172 *
173  IF( n.EQ.0 ) THEN
174  scond = one
175  amax = zero
176  RETURN
177  END IF
178
179  base = slamch( 'B' )
180  tmp = -0.5 / log( base )
181 *
182 * Find the minimum and maximum diagonal elements.
183 *
184  s( 1 ) = a( 1, 1 )
185  smin = s( 1 )
186  amax = s( 1 )
187  DO 10 i = 2, n
188  s( i ) = a( i, i )
189  smin = min( smin, s( i ) )
190  amax = max( amax, s( i ) )
191  10 CONTINUE
192 *
193  IF( smin.LE.zero ) THEN
194 *
195 * Find the first non-positive diagonal element and return.
196 *
197  DO 20 i = 1, n
198  IF( s( i ).LE.zero ) THEN
199  info = i
200  RETURN
201  END IF
202  20 CONTINUE
203  ELSE
204 *
205 * Set the scale factors to the reciprocals
206 * of the diagonal elements.
207 *
208  DO 30 i = 1, n
209  s( i ) = base ** int( tmp * log( s( i ) ) )
210  30 CONTINUE
211 *
212 * Compute SCOND = min(S(I)) / max(S(I)).
213 *
214  scond = sqrt( smin ) / sqrt( amax )
215  END IF
216 *
217  RETURN
218 *
219 * End of SPOEQUB
220 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
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