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

 subroutine sppequ ( character UPLO, integer N, real, dimension( * ) AP, real, dimension( * ) S, real SCOND, real AMAX, integer INFO )

SPPEQU

Purpose:
``` SPPEQU computes row and column scalings intended to equilibrate a
symmetric positive definite matrix A in packed storage 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.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in] AP ``` AP is REAL array, dimension (N*(N+1)/2) The upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=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.```

Definition at line 115 of file sppequ.f.

116*
117* -- LAPACK computational routine --
118* -- LAPACK is a software package provided by Univ. of Tennessee, --
119* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
120*
121* .. Scalar Arguments ..
122 CHARACTER UPLO
123 INTEGER INFO, N
124 REAL AMAX, SCOND
125* ..
126* .. Array Arguments ..
127 REAL AP( * ), S( * )
128* ..
129*
130* =====================================================================
131*
132* .. Parameters ..
133 REAL ONE, ZERO
134 parameter( one = 1.0e+0, zero = 0.0e+0 )
135* ..
136* .. Local Scalars ..
137 LOGICAL UPPER
138 INTEGER I, JJ
139 REAL SMIN
140* ..
141* .. External Functions ..
142 LOGICAL LSAME
143 EXTERNAL lsame
144* ..
145* .. External Subroutines ..
146 EXTERNAL xerbla
147* ..
148* .. Intrinsic Functions ..
149 INTRINSIC max, min, sqrt
150* ..
151* .. Executable Statements ..
152*
153* Test the input parameters.
154*
155 info = 0
156 upper = lsame( uplo, 'U' )
157 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
158 info = -1
159 ELSE IF( n.LT.0 ) THEN
160 info = -2
161 END IF
162 IF( info.NE.0 ) THEN
163 CALL xerbla( 'SPPEQU', -info )
164 RETURN
165 END IF
166*
167* Quick return if possible
168*
169 IF( n.EQ.0 ) THEN
170 scond = one
171 amax = zero
172 RETURN
173 END IF
174*
175* Initialize SMIN and AMAX.
176*
177 s( 1 ) = ap( 1 )
178 smin = s( 1 )
179 amax = s( 1 )
180*
181 IF( upper ) THEN
182*
183* UPLO = 'U': Upper triangle of A is stored.
184* Find the minimum and maximum diagonal elements.
185*
186 jj = 1
187 DO 10 i = 2, n
188 jj = jj + i
189 s( i ) = ap( jj )
190 smin = min( smin, s( i ) )
191 amax = max( amax, s( i ) )
192 10 CONTINUE
193*
194 ELSE
195*
196* UPLO = 'L': Lower triangle of A is stored.
197* Find the minimum and maximum diagonal elements.
198*
199 jj = 1
200 DO 20 i = 2, n
201 jj = jj + n - i + 2
202 s( i ) = ap( jj )
203 smin = min( smin, s( i ) )
204 amax = max( amax, s( i ) )
205 20 CONTINUE
206 END IF
207*
208 IF( smin.LE.zero ) THEN
209*
210* Find the first non-positive diagonal element and return.
211*
212 DO 30 i = 1, n
213 IF( s( i ).LE.zero ) THEN
214 info = i
215 RETURN
216 END IF
217 30 CONTINUE
218 ELSE
219*
220* Set the scale factors to the reciprocals
221* of the diagonal elements.
222*
223 DO 40 i = 1, n
224 s( i ) = one / sqrt( s( i ) )
225 40 CONTINUE
226*
227* Compute SCOND = min(S(I)) / max(S(I))
228*
229 scond = sqrt( smin ) / sqrt( amax )
230 END IF
231 RETURN
232*
233* End of SPPEQU
234*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
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