LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ 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

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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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
December 2016

Definition at line 132 of file zpbequ.f.

132 *
133 * -- LAPACK computational routine (version 3.7.0) --
134 * -- LAPACK is a software package provided by Univ. of Tennessee, --
135 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
136 * December 2016
137 *
138 * .. Scalar Arguments ..
139  CHARACTER uplo
140  INTEGER info, kd, ldab, n
141  DOUBLE PRECISION amax, scond
142 * ..
143 * .. Array Arguments ..
144  DOUBLE PRECISION s( * )
145  COMPLEX*16 ab( ldab, * )
146 * ..
147 *
148 * =====================================================================
149 *
150 * .. Parameters ..
151  DOUBLE PRECISION zero, one
152  parameter( zero = 0.0d+0, one = 1.0d+0 )
153 * ..
154 * .. Local Scalars ..
155  LOGICAL upper
156  INTEGER i, j
157  DOUBLE PRECISION smin
158 * ..
159 * .. External Functions ..
160  LOGICAL lsame
161  EXTERNAL lsame
162 * ..
163 * .. External Subroutines ..
164  EXTERNAL xerbla
165 * ..
166 * .. Intrinsic Functions ..
167  INTRINSIC dble, max, min, sqrt
168 * ..
169 * .. Executable Statements ..
170 *
171 * Test the input parameters.
172 *
173  info = 0
174  upper = lsame( uplo, 'U' )
175  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
176  info = -1
177  ELSE IF( n.LT.0 ) THEN
178  info = -2
179  ELSE IF( kd.LT.0 ) THEN
180  info = -3
181  ELSE IF( ldab.LT.kd+1 ) THEN
182  info = -5
183  END IF
184  IF( info.NE.0 ) THEN
185  CALL xerbla( 'ZPBEQU', -info )
186  RETURN
187  END IF
188 *
189 * Quick return if possible
190 *
191  IF( n.EQ.0 ) THEN
192  scond = one
193  amax = zero
194  RETURN
195  END IF
196 *
197  IF( upper ) THEN
198  j = kd + 1
199  ELSE
200  j = 1
201  END IF
202 *
203 * Initialize SMIN and AMAX.
204 *
205  s( 1 ) = dble( ab( j, 1 ) )
206  smin = s( 1 )
207  amax = s( 1 )
208 *
209 * Find the minimum and maximum diagonal elements.
210 *
211  DO 10 i = 2, n
212  s( i ) = dble( ab( j, i ) )
213  smin = min( smin, s( i ) )
214  amax = max( amax, s( i ) )
215  10 CONTINUE
216 *
217  IF( smin.LE.zero ) THEN
218 *
219 * Find the first non-positive diagonal element and return.
220 *
221  DO 20 i = 1, n
222  IF( s( i ).LE.zero ) THEN
223  info = i
224  RETURN
225  END IF
226  20 CONTINUE
227  ELSE
228 *
229 * Set the scale factors to the reciprocals
230 * of the diagonal elements.
231 *
232  DO 30 i = 1, n
233  s( i ) = one / sqrt( s( i ) )
234  30 CONTINUE
235 *
236 * Compute SCOND = min(S(I)) / max(S(I))
237 *
238  scond = sqrt( smin ) / sqrt( amax )
239  END IF
240  RETURN
241 *
242 * End of ZPBEQU
243 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
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