LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ dlagtf()

subroutine dlagtf ( integer  N,
double precision, dimension( * )  A,
double precision  LAMBDA,
double precision, dimension( * )  B,
double precision, dimension( * )  C,
double precision  TOL,
double precision, dimension( * )  D,
integer, dimension( * )  IN,
integer  INFO 
)

DLAGTF computes an LU factorization of a matrix T-λI, where T is a general tridiagonal matrix, and λ a scalar, using partial pivoting with row interchanges.

Download DLAGTF + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 DLAGTF factorizes the matrix (T - lambda*I), where T is an n by n
 tridiagonal matrix and lambda is a scalar, as

    T - lambda*I = PLU,

 where P is a permutation matrix, L is a unit lower tridiagonal matrix
 with at most one non-zero sub-diagonal elements per column and U is
 an upper triangular matrix with at most two non-zero super-diagonal
 elements per column.

 The factorization is obtained by Gaussian elimination with partial
 pivoting and implicit row scaling.

 The parameter LAMBDA is included in the routine so that DLAGTF may
 be used, in conjunction with DLAGTS, to obtain eigenvectors of T by
 inverse iteration.
Parameters
[in]N
          N is INTEGER
          The order of the matrix T.
[in,out]A
          A is DOUBLE PRECISION array, dimension (N)
          On entry, A must contain the diagonal elements of T.

          On exit, A is overwritten by the n diagonal elements of the
          upper triangular matrix U of the factorization of T.
[in]LAMBDA
          LAMBDA is DOUBLE PRECISION
          On entry, the scalar lambda.
[in,out]B
          B is DOUBLE PRECISION array, dimension (N-1)
          On entry, B must contain the (n-1) super-diagonal elements of
          T.

          On exit, B is overwritten by the (n-1) super-diagonal
          elements of the matrix U of the factorization of T.
[in,out]C
          C is DOUBLE PRECISION array, dimension (N-1)
          On entry, C must contain the (n-1) sub-diagonal elements of
          T.

          On exit, C is overwritten by the (n-1) sub-diagonal elements
          of the matrix L of the factorization of T.
[in]TOL
          TOL is DOUBLE PRECISION
          On entry, a relative tolerance used to indicate whether or
          not the matrix (T - lambda*I) is nearly singular. TOL should
          normally be chose as approximately the largest relative error
          in the elements of T. For example, if the elements of T are
          correct to about 4 significant figures, then TOL should be
          set to about 5*10**(-4). If TOL is supplied as less than eps,
          where eps is the relative machine precision, then the value
          eps is used in place of TOL.
[out]D
          D is DOUBLE PRECISION array, dimension (N-2)
          On exit, D is overwritten by the (n-2) second super-diagonal
          elements of the matrix U of the factorization of T.
[out]IN
          IN is INTEGER array, dimension (N)
          On exit, IN contains details of the permutation matrix P. If
          an interchange occurred at the kth step of the elimination,
          then IN(k) = 1, otherwise IN(k) = 0. The element IN(n)
          returns the smallest positive integer j such that

             abs( u(j,j) ) <= norm( (T - lambda*I)(j) )*TOL,

          where norm( A(j) ) denotes the sum of the absolute values of
          the jth row of the matrix A. If no such j exists then IN(n)
          is returned as zero. If IN(n) is returned as positive, then a
          diagonal element of U is small, indicating that
          (T - lambda*I) is singular or nearly singular,
[out]INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -k, the kth argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 155 of file dlagtf.f.

156 *
157 * -- LAPACK computational routine --
158 * -- LAPACK is a software package provided by Univ. of Tennessee, --
159 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
160 *
161 * .. Scalar Arguments ..
162  INTEGER INFO, N
163  DOUBLE PRECISION LAMBDA, TOL
164 * ..
165 * .. Array Arguments ..
166  INTEGER IN( * )
167  DOUBLE PRECISION A( * ), B( * ), C( * ), D( * )
168 * ..
169 *
170 * =====================================================================
171 *
172 * .. Parameters ..
173  DOUBLE PRECISION ZERO
174  parameter( zero = 0.0d+0 )
175 * ..
176 * .. Local Scalars ..
177  INTEGER K
178  DOUBLE PRECISION EPS, MULT, PIV1, PIV2, SCALE1, SCALE2, TEMP, TL
179 * ..
180 * .. Intrinsic Functions ..
181  INTRINSIC abs, max
182 * ..
183 * .. External Functions ..
184  DOUBLE PRECISION DLAMCH
185  EXTERNAL dlamch
186 * ..
187 * .. External Subroutines ..
188  EXTERNAL xerbla
189 * ..
190 * .. Executable Statements ..
191 *
192  info = 0
193  IF( n.LT.0 ) THEN
194  info = -1
195  CALL xerbla( 'DLAGTF', -info )
196  RETURN
197  END IF
198 *
199  IF( n.EQ.0 )
200  $ RETURN
201 *
202  a( 1 ) = a( 1 ) - lambda
203  in( n ) = 0
204  IF( n.EQ.1 ) THEN
205  IF( a( 1 ).EQ.zero )
206  $ in( 1 ) = 1
207  RETURN
208  END IF
209 *
210  eps = dlamch( 'Epsilon' )
211 *
212  tl = max( tol, eps )
213  scale1 = abs( a( 1 ) ) + abs( b( 1 ) )
214  DO 10 k = 1, n - 1
215  a( k+1 ) = a( k+1 ) - lambda
216  scale2 = abs( c( k ) ) + abs( a( k+1 ) )
217  IF( k.LT.( n-1 ) )
218  $ scale2 = scale2 + abs( b( k+1 ) )
219  IF( a( k ).EQ.zero ) THEN
220  piv1 = zero
221  ELSE
222  piv1 = abs( a( k ) ) / scale1
223  END IF
224  IF( c( k ).EQ.zero ) THEN
225  in( k ) = 0
226  piv2 = zero
227  scale1 = scale2
228  IF( k.LT.( n-1 ) )
229  $ d( k ) = zero
230  ELSE
231  piv2 = abs( c( k ) ) / scale2
232  IF( piv2.LE.piv1 ) THEN
233  in( k ) = 0
234  scale1 = scale2
235  c( k ) = c( k ) / a( k )
236  a( k+1 ) = a( k+1 ) - c( k )*b( k )
237  IF( k.LT.( n-1 ) )
238  $ d( k ) = zero
239  ELSE
240  in( k ) = 1
241  mult = a( k ) / c( k )
242  a( k ) = c( k )
243  temp = a( k+1 )
244  a( k+1 ) = b( k ) - mult*temp
245  IF( k.LT.( n-1 ) ) THEN
246  d( k ) = b( k+1 )
247  b( k+1 ) = -mult*d( k )
248  END IF
249  b( k ) = temp
250  c( k ) = mult
251  END IF
252  END IF
253  IF( ( max( piv1, piv2 ).LE.tl ) .AND. ( in( n ).EQ.0 ) )
254  $ in( n ) = k
255  10 CONTINUE
256  IF( ( abs( a( n ) ).LE.scale1*tl ) .AND. ( in( n ).EQ.0 ) )
257  $ in( n ) = n
258 *
259  RETURN
260 *
261 * End of DLAGTF
262 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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