 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ spbequ()

 subroutine spbequ ( character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) S, real SCOND, real AMAX, integer INFO )

SPBEQU

Purpose:
``` SPBEQU computes row and column scalings intended to equilibrate a
symmetric positive definite band 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.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangular of A is stored; = 'L': Lower triangular of A is stored.``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in] KD ``` KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.``` [in] AB ``` AB is REAL array, dimension (LDAB,N) The upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).``` [in] LDAB ``` LDAB is INTEGER The leading dimension of the array A. LDAB >= KD+1.``` [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 131 of file spbequ.f.

131 *
132 * -- LAPACK computational routine (version 3.7.0) --
133 * -- LAPACK is a software package provided by Univ. of Tennessee, --
134 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
135 * December 2016
136 *
137 * .. Scalar Arguments ..
138  CHARACTER uplo
139  INTEGER info, kd, ldab, n
140  REAL amax, scond
141 * ..
142 * .. Array Arguments ..
143  REAL ab( ldab, * ), s( * )
144 * ..
145 *
146 * =====================================================================
147 *
148 * .. Parameters ..
149  REAL zero, one
150  parameter( zero = 0.0e+0, one = 1.0e+0 )
151 * ..
152 * .. Local Scalars ..
153  LOGICAL upper
154  INTEGER i, j
155  REAL smin
156 * ..
157 * .. External Functions ..
158  LOGICAL lsame
159  EXTERNAL lsame
160 * ..
161 * .. External Subroutines ..
162  EXTERNAL xerbla
163 * ..
164 * .. Intrinsic Functions ..
165  INTRINSIC max, min, sqrt
166 * ..
167 * .. Executable Statements ..
168 *
169 * Test the input parameters.
170 *
171  info = 0
172  upper = lsame( uplo, 'U' )
173  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
174  info = -1
175  ELSE IF( n.LT.0 ) THEN
176  info = -2
177  ELSE IF( kd.LT.0 ) THEN
178  info = -3
179  ELSE IF( ldab.LT.kd+1 ) THEN
180  info = -5
181  END IF
182  IF( info.NE.0 ) THEN
183  CALL xerbla( 'SPBEQU', -info )
184  RETURN
185  END IF
186 *
187 * Quick return if possible
188 *
189  IF( n.EQ.0 ) THEN
190  scond = one
191  amax = zero
192  RETURN
193  END IF
194 *
195  IF( upper ) THEN
196  j = kd + 1
197  ELSE
198  j = 1
199  END IF
200 *
201 * Initialize SMIN and AMAX.
202 *
203  s( 1 ) = ab( j, 1 )
204  smin = s( 1 )
205  amax = s( 1 )
206 *
207 * Find the minimum and maximum diagonal elements.
208 *
209  DO 10 i = 2, n
210  s( i ) = ab( j, i )
211  smin = min( smin, s( i ) )
212  amax = max( amax, s( i ) )
213  10 CONTINUE
214 *
215  IF( smin.LE.zero ) THEN
216 *
217 * Find the first non-positive diagonal element and return.
218 *
219  DO 20 i = 1, n
220  IF( s( i ).LE.zero ) THEN
221  info = i
222  RETURN
223  END IF
224  20 CONTINUE
225  ELSE
226 *
227 * Set the scale factors to the reciprocals
228 * of the diagonal elements.
229 *
230  DO 30 i = 1, n
231  s( i ) = one / sqrt( s( i ) )
232  30 CONTINUE
233 *
234 * Compute SCOND = min(S(I)) / max(S(I))
235 *
236  scond = sqrt( smin ) / sqrt( amax )
237  END IF
238  RETURN
239 *
240 * End of SPBEQU
241 *
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
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