LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ 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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