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

 subroutine dpbequ ( character UPLO, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, integer INFO )

DPBEQU

Purpose:
``` DPBEQU 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 DOUBLE PRECISION 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 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.```

Definition at line 128 of file dpbequ.f.

129*
130* -- LAPACK computational routine --
131* -- LAPACK is a software package provided by Univ. of Tennessee, --
132* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
133*
134* .. Scalar Arguments ..
135 CHARACTER UPLO
136 INTEGER INFO, KD, LDAB, N
137 DOUBLE PRECISION AMAX, SCOND
138* ..
139* .. Array Arguments ..
140 DOUBLE PRECISION AB( LDAB, * ), S( * )
141* ..
142*
143* =====================================================================
144*
145* .. Parameters ..
146 DOUBLE PRECISION ZERO, ONE
147 parameter( zero = 0.0d+0, one = 1.0d+0 )
148* ..
149* .. Local Scalars ..
150 LOGICAL UPPER
151 INTEGER I, J
152 DOUBLE PRECISION SMIN
153* ..
154* .. External Functions ..
155 LOGICAL LSAME
156 EXTERNAL lsame
157* ..
158* .. External Subroutines ..
159 EXTERNAL xerbla
160* ..
161* .. Intrinsic Functions ..
162 INTRINSIC max, min, sqrt
163* ..
164* .. Executable Statements ..
165*
166* Test the input parameters.
167*
168 info = 0
169 upper = lsame( uplo, 'U' )
170 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
171 info = -1
172 ELSE IF( n.LT.0 ) THEN
173 info = -2
174 ELSE IF( kd.LT.0 ) THEN
175 info = -3
176 ELSE IF( ldab.LT.kd+1 ) THEN
177 info = -5
178 END IF
179 IF( info.NE.0 ) THEN
180 CALL xerbla( 'DPBEQU', -info )
181 RETURN
182 END IF
183*
184* Quick return if possible
185*
186 IF( n.EQ.0 ) THEN
187 scond = one
188 amax = zero
189 RETURN
190 END IF
191*
192 IF( upper ) THEN
193 j = kd + 1
194 ELSE
195 j = 1
196 END IF
197*
198* Initialize SMIN and AMAX.
199*
200 s( 1 ) = ab( j, 1 )
201 smin = s( 1 )
202 amax = s( 1 )
203*
204* Find the minimum and maximum diagonal elements.
205*
206 DO 10 i = 2, n
207 s( i ) = ab( j, i )
208 smin = min( smin, s( i ) )
209 amax = max( amax, s( i ) )
210 10 CONTINUE
211*
212 IF( smin.LE.zero ) THEN
213*
214* Find the first non-positive diagonal element and return.
215*
216 DO 20 i = 1, n
217 IF( s( i ).LE.zero ) THEN
218 info = i
219 RETURN
220 END IF
221 20 CONTINUE
222 ELSE
223*
224* Set the scale factors to the reciprocals
225* of the diagonal elements.
226*
227 DO 30 i = 1, n
228 s( i ) = one / sqrt( s( i ) )
229 30 CONTINUE
230*
231* Compute SCOND = min(S(I)) / max(S(I))
232*
233 scond = sqrt( smin ) / sqrt( amax )
234 END IF
235 RETURN
236*
237* End of DPBEQU
238*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
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