LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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zpbtf2.f
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1*> \brief \b ZPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
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16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE ZPBTF2( UPLO, N, KD, AB, LDAB, INFO )
22*
23* .. Scalar Arguments ..
24* CHARACTER UPLO
25* INTEGER INFO, KD, LDAB, N
26* ..
27* .. Array Arguments ..
28* COMPLEX*16 AB( LDAB, * )
29* ..
30*
31*
32*> \par Purpose:
33* =============
34*>
35*> \verbatim
36*>
37*> ZPBTF2 computes the Cholesky factorization of a complex Hermitian
38*> positive definite band matrix A.
39*>
40*> The factorization has the form
41*> A = U**H * U , if UPLO = 'U', or
42*> A = L * L**H, if UPLO = 'L',
43*> where U is an upper triangular matrix, U**H is the conjugate transpose
44*> of U, and L is lower triangular.
45*>
46*> This is the unblocked version of the algorithm, calling Level 2 BLAS.
47*> \endverbatim
48*
49* Arguments:
50* ==========
51*
52*> \param[in] UPLO
53*> \verbatim
54*> UPLO is CHARACTER*1
55*> Specifies whether the upper or lower triangular part of the
56*> Hermitian matrix A is stored:
57*> = 'U': Upper triangular
58*> = 'L': Lower triangular
59*> \endverbatim
60*>
61*> \param[in] N
62*> \verbatim
63*> N is INTEGER
64*> The order of the matrix A. N >= 0.
65*> \endverbatim
66*>
67*> \param[in] KD
68*> \verbatim
69*> KD is INTEGER
70*> The number of super-diagonals of the matrix A if UPLO = 'U',
71*> or the number of sub-diagonals if UPLO = 'L'. KD >= 0.
72*> \endverbatim
73*>
74*> \param[in,out] AB
75*> \verbatim
76*> AB is COMPLEX*16 array, dimension (LDAB,N)
77*> On entry, the upper or lower triangle of the Hermitian band
78*> matrix A, stored in the first KD+1 rows of the array. The
79*> j-th column of A is stored in the j-th column of the array AB
80*> as follows:
81*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
82*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
83*>
84*> On exit, if INFO = 0, the triangular factor U or L from the
85*> Cholesky factorization A = U**H *U or A = L*L**H of the band
86*> matrix A, in the same storage format as A.
87*> \endverbatim
88*>
89*> \param[in] LDAB
90*> \verbatim
91*> LDAB is INTEGER
92*> The leading dimension of the array AB. LDAB >= KD+1.
93*> \endverbatim
94*>
95*> \param[out] INFO
96*> \verbatim
97*> INFO is INTEGER
98*> = 0: successful exit
99*> < 0: if INFO = -k, the k-th argument had an illegal value
100*> > 0: if INFO = k, the leading minor of order k is not
101*> positive definite, and the factorization could not be
102*> completed.
103*> \endverbatim
104*
105* Authors:
106* ========
107*
108*> \author Univ. of Tennessee
109*> \author Univ. of California Berkeley
110*> \author Univ. of Colorado Denver
111*> \author NAG Ltd.
112*
113*> \ingroup complex16OTHERcomputational
114*
115*> \par Further Details:
116* =====================
117*>
118*> \verbatim
119*>
120*> The band storage scheme is illustrated by the following example, when
121*> N = 6, KD = 2, and UPLO = 'U':
122*>
123*> On entry: On exit:
124*>
125*> * * a13 a24 a35 a46 * * u13 u24 u35 u46
126*> * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
127*> a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
128*>
129*> Similarly, if UPLO = 'L' the format of A is as follows:
130*>
131*> On entry: On exit:
132*>
133*> a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
134*> a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
135*> a31 a42 a53 a64 * * l31 l42 l53 l64 * *
136*>
137*> Array elements marked * are not used by the routine.
138*> \endverbatim
139*>
140* =====================================================================
141 SUBROUTINE zpbtf2( UPLO, N, KD, AB, LDAB, INFO )
142*
143* -- LAPACK computational routine --
144* -- LAPACK is a software package provided by Univ. of Tennessee, --
145* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
146*
147* .. Scalar Arguments ..
148 CHARACTER UPLO
149 INTEGER INFO, KD, LDAB, N
150* ..
151* .. Array Arguments ..
152 COMPLEX*16 AB( LDAB, * )
153* ..
154*
155* =====================================================================
156*
157* .. Parameters ..
158 DOUBLE PRECISION ONE, ZERO
159 parameter( one = 1.0d+0, zero = 0.0d+0 )
160* ..
161* .. Local Scalars ..
162 LOGICAL UPPER
163 INTEGER J, KLD, KN
164 DOUBLE PRECISION AJJ
165* ..
166* .. External Functions ..
167 LOGICAL LSAME
168 EXTERNAL lsame
169* ..
170* .. External Subroutines ..
171 EXTERNAL xerbla, zdscal, zher, zlacgv
172* ..
173* .. Intrinsic Functions ..
174 INTRINSIC dble, max, min, sqrt
175* ..
176* .. Executable Statements ..
177*
178* Test the input parameters.
179*
180 info = 0
181 upper = lsame( uplo, 'U' )
182 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
183 info = -1
184 ELSE IF( n.LT.0 ) THEN
185 info = -2
186 ELSE IF( kd.LT.0 ) THEN
187 info = -3
188 ELSE IF( ldab.LT.kd+1 ) THEN
189 info = -5
190 END IF
191 IF( info.NE.0 ) THEN
192 CALL xerbla( 'ZPBTF2', -info )
193 RETURN
194 END IF
195*
196* Quick return if possible
197*
198 IF( n.EQ.0 )
199 \$ RETURN
200*
201 kld = max( 1, ldab-1 )
202*
203 IF( upper ) THEN
204*
205* Compute the Cholesky factorization A = U**H * U.
206*
207 DO 10 j = 1, n
208*
209* Compute U(J,J) and test for non-positive-definiteness.
210*
211 ajj = dble( ab( kd+1, j ) )
212 IF( ajj.LE.zero ) THEN
213 ab( kd+1, j ) = ajj
214 GO TO 30
215 END IF
216 ajj = sqrt( ajj )
217 ab( kd+1, j ) = ajj
218*
219* Compute elements J+1:J+KN of row J and update the
220* trailing submatrix within the band.
221*
222 kn = min( kd, n-j )
223 IF( kn.GT.0 ) THEN
224 CALL zdscal( kn, one / ajj, ab( kd, j+1 ), kld )
225 CALL zlacgv( kn, ab( kd, j+1 ), kld )
226 CALL zher( 'Upper', kn, -one, ab( kd, j+1 ), kld,
227 \$ ab( kd+1, j+1 ), kld )
228 CALL zlacgv( kn, ab( kd, j+1 ), kld )
229 END IF
230 10 CONTINUE
231 ELSE
232*
233* Compute the Cholesky factorization A = L*L**H.
234*
235 DO 20 j = 1, n
236*
237* Compute L(J,J) and test for non-positive-definiteness.
238*
239 ajj = dble( ab( 1, j ) )
240 IF( ajj.LE.zero ) THEN
241 ab( 1, j ) = ajj
242 GO TO 30
243 END IF
244 ajj = sqrt( ajj )
245 ab( 1, j ) = ajj
246*
247* Compute elements J+1:J+KN of column J and update the
248* trailing submatrix within the band.
249*
250 kn = min( kd, n-j )
251 IF( kn.GT.0 ) THEN
252 CALL zdscal( kn, one / ajj, ab( 2, j ), 1 )
253 CALL zher( 'Lower', kn, -one, ab( 2, j ), 1,
254 \$ ab( 1, j+1 ), kld )
255 END IF
256 20 CONTINUE
257 END IF
258 RETURN
259*
260 30 CONTINUE
261 info = j
262 RETURN
263*
264* End of ZPBTF2
265*
266 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:78
subroutine zher(UPLO, N, ALPHA, X, INCX, A, LDA)
ZHER
Definition: zher.f:135
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:74
subroutine zpbtf2(UPLO, N, KD, AB, LDAB, INFO)
ZPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (un...
Definition: zpbtf2.f:142