 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ slagtf()

 subroutine slagtf ( integer N, real, dimension( * ) A, real LAMBDA, real, dimension( * ) B, real, dimension( * ) C, real TOL, real, dimension( * ) D, integer, dimension( * ) IN, integer INFO )

SLAGTF 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.

Purpose:
``` SLAGTF 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 SLAGTF may
be used, in conjunction with SLAGTS, 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 REAL 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 REAL On entry, the scalar lambda.``` [in,out] B ``` B is REAL 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 REAL 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 REAL 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 REAL 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```

Definition at line 155 of file slagtf.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  REAL LAMBDA, TOL
164 * ..
165 * .. Array Arguments ..
166  INTEGER IN( * )
167  REAL A( * ), B( * ), C( * ), D( * )
168 * ..
169 *
170 * =====================================================================
171 *
172 * .. Parameters ..
173  REAL ZERO
174  parameter( zero = 0.0e+0 )
175 * ..
176 * .. Local Scalars ..
177  INTEGER K
178  REAL EPS, MULT, PIV1, PIV2, SCALE1, SCALE2, TEMP, TL
179 * ..
180 * .. Intrinsic Functions ..
181  INTRINSIC abs, max
182 * ..
183 * .. External Functions ..
184  REAL SLAMCH
185  EXTERNAL slamch
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( 'SLAGTF', -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 = slamch( '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 SLAGTF
262 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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