LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ 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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