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

 subroutine zpbequ ( character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, integer INFO )

ZPBEQU

Purpose:
``` ZPBEQU computes row and column scalings intended to equilibrate a
Hermitian 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 COMPLEX*16 array, dimension (LDAB,N) The upper or lower triangle of the Hermitian 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 129 of file zpbequ.f.

130*
131* -- LAPACK computational routine --
132* -- LAPACK is a software package provided by Univ. of Tennessee, --
133* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
134*
135* .. Scalar Arguments ..
136 CHARACTER UPLO
137 INTEGER INFO, KD, LDAB, N
138 DOUBLE PRECISION AMAX, SCOND
139* ..
140* .. Array Arguments ..
141 DOUBLE PRECISION S( * )
142 COMPLEX*16 AB( LDAB, * )
143* ..
144*
145* =====================================================================
146*
147* .. Parameters ..
148 DOUBLE PRECISION ZERO, ONE
149 parameter( zero = 0.0d+0, one = 1.0d+0 )
150* ..
151* .. Local Scalars ..
152 LOGICAL UPPER
153 INTEGER I, J
154 DOUBLE PRECISION SMIN
155* ..
156* .. External Functions ..
157 LOGICAL LSAME
158 EXTERNAL lsame
159* ..
160* .. External Subroutines ..
161 EXTERNAL xerbla
162* ..
163* .. Intrinsic Functions ..
164 INTRINSIC dble, max, min, sqrt
165* ..
166* .. Executable Statements ..
167*
168* Test the input parameters.
169*
170 info = 0
171 upper = lsame( uplo, 'U' )
172 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
173 info = -1
174 ELSE IF( n.LT.0 ) THEN
175 info = -2
176 ELSE IF( kd.LT.0 ) THEN
177 info = -3
178 ELSE IF( ldab.LT.kd+1 ) THEN
179 info = -5
180 END IF
181 IF( info.NE.0 ) THEN
182 CALL xerbla( 'ZPBEQU', -info )
183 RETURN
184 END IF
185*
186* Quick return if possible
187*
188 IF( n.EQ.0 ) THEN
189 scond = one
190 amax = zero
191 RETURN
192 END IF
193*
194 IF( upper ) THEN
195 j = kd + 1
196 ELSE
197 j = 1
198 END IF
199*
200* Initialize SMIN and AMAX.
201*
202 s( 1 ) = dble( ab( j, 1 ) )
203 smin = s( 1 )
204 amax = s( 1 )
205*
206* Find the minimum and maximum diagonal elements.
207*
208 DO 10 i = 2, n
209 s( i ) = dble( ab( j, i ) )
210 smin = min( smin, s( i ) )
211 amax = max( amax, s( i ) )
212 10 CONTINUE
213*
214 IF( smin.LE.zero ) THEN
215*
216* Find the first non-positive diagonal element and return.
217*
218 DO 20 i = 1, n
219 IF( s( i ).LE.zero ) THEN
220 info = i
221 RETURN
222 END IF
223 20 CONTINUE
224 ELSE
225*
226* Set the scale factors to the reciprocals
227* of the diagonal elements.
228*
229 DO 30 i = 1, n
230 s( i ) = one / sqrt( s( i ) )
231 30 CONTINUE
232*
233* Compute SCOND = min(S(I)) / max(S(I))
234*
235 scond = sqrt( smin ) / sqrt( amax )
236 END IF
237 RETURN
238*
239* End of ZPBEQU
240*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
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