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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b CPBTRF 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download CPBTRF + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cpbtrf.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cpbtrf.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cpbtrf.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE CPBTRF( UPLO, N, KD, AB, LDAB, INFO ) 00022 * 00023 * .. Scalar Arguments .. 00024 * CHARACTER UPLO 00025 * INTEGER INFO, KD, LDAB, N 00026 * .. 00027 * .. Array Arguments .. 00028 * COMPLEX AB( LDAB, * ) 00029 * .. 00030 * 00031 * 00032 *> \par Purpose: 00033 * ============= 00034 *> 00035 *> \verbatim 00036 *> 00037 *> CPBTRF computes the Cholesky factorization of a complex Hermitian 00038 *> positive definite band matrix A. 00039 *> 00040 *> The factorization has the form 00041 *> A = U**H * U, if UPLO = 'U', or 00042 *> A = L * L**H, if UPLO = 'L', 00043 *> where U is an upper triangular matrix and L is lower triangular. 00044 *> \endverbatim 00045 * 00046 * Arguments: 00047 * ========== 00048 * 00049 *> \param[in] UPLO 00050 *> \verbatim 00051 *> UPLO is CHARACTER*1 00052 *> = 'U': Upper triangle of A is stored; 00053 *> = 'L': Lower triangle of A is stored. 00054 *> \endverbatim 00055 *> 00056 *> \param[in] N 00057 *> \verbatim 00058 *> N is INTEGER 00059 *> The order of the matrix A. N >= 0. 00060 *> \endverbatim 00061 *> 00062 *> \param[in] KD 00063 *> \verbatim 00064 *> KD is INTEGER 00065 *> The number of superdiagonals of the matrix A if UPLO = 'U', 00066 *> or the number of subdiagonals if UPLO = 'L'. KD >= 0. 00067 *> \endverbatim 00068 *> 00069 *> \param[in,out] AB 00070 *> \verbatim 00071 *> AB is COMPLEX array, dimension (LDAB,N) 00072 *> On entry, the upper or lower triangle of the Hermitian band 00073 *> matrix A, stored in the first KD+1 rows of the array. The 00074 *> j-th column of A is stored in the j-th column of the array AB 00075 *> as follows: 00076 *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; 00077 *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). 00078 *> 00079 *> On exit, if INFO = 0, the triangular factor U or L from the 00080 *> Cholesky factorization A = U**H*U or A = L*L**H of the band 00081 *> matrix A, in the same storage format as A. 00082 *> \endverbatim 00083 *> 00084 *> \param[in] LDAB 00085 *> \verbatim 00086 *> LDAB is INTEGER 00087 *> The leading dimension of the array AB. LDAB >= KD+1. 00088 *> \endverbatim 00089 *> 00090 *> \param[out] INFO 00091 *> \verbatim 00092 *> INFO is INTEGER 00093 *> = 0: successful exit 00094 *> < 0: if INFO = -i, the i-th argument had an illegal value 00095 *> > 0: if INFO = i, the leading minor of order i is not 00096 *> positive definite, and the factorization could not be 00097 *> completed. 00098 *> \endverbatim 00099 * 00100 * Authors: 00101 * ======== 00102 * 00103 *> \author Univ. of Tennessee 00104 *> \author Univ. of California Berkeley 00105 *> \author Univ. of Colorado Denver 00106 *> \author NAG Ltd. 00107 * 00108 *> \date November 2011 00109 * 00110 *> \ingroup complexOTHERcomputational 00111 * 00112 *> \par Further Details: 00113 * ===================== 00114 *> 00115 *> \verbatim 00116 *> 00117 *> The band storage scheme is illustrated by the following example, when 00118 *> N = 6, KD = 2, and UPLO = 'U': 00119 *> 00120 *> On entry: On exit: 00121 *> 00122 *> * * a13 a24 a35 a46 * * u13 u24 u35 u46 00123 *> * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 00124 *> a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 00125 *> 00126 *> Similarly, if UPLO = 'L' the format of A is as follows: 00127 *> 00128 *> On entry: On exit: 00129 *> 00130 *> a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66 00131 *> a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 * 00132 *> a31 a42 a53 a64 * * l31 l42 l53 l64 * * 00133 *> 00134 *> Array elements marked * are not used by the routine. 00135 *> \endverbatim 00136 * 00137 *> \par Contributors: 00138 * ================== 00139 *> 00140 *> Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989 00141 * 00142 * ===================================================================== 00143 SUBROUTINE CPBTRF( UPLO, N, KD, AB, LDAB, INFO ) 00144 * 00145 * -- LAPACK computational routine (version 3.4.0) -- 00146 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00147 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00148 * November 2011 00149 * 00150 * .. Scalar Arguments .. 00151 CHARACTER UPLO 00152 INTEGER INFO, KD, LDAB, N 00153 * .. 00154 * .. Array Arguments .. 00155 COMPLEX AB( LDAB, * ) 00156 * .. 00157 * 00158 * ===================================================================== 00159 * 00160 * .. Parameters .. 00161 REAL ONE, ZERO 00162 PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 ) 00163 COMPLEX CONE 00164 PARAMETER ( CONE = ( 1.0E+0, 0.0E+0 ) ) 00165 INTEGER NBMAX, LDWORK 00166 PARAMETER ( NBMAX = 32, LDWORK = NBMAX+1 ) 00167 * .. 00168 * .. Local Scalars .. 00169 INTEGER I, I2, I3, IB, II, J, JJ, NB 00170 * .. 00171 * .. Local Arrays .. 00172 COMPLEX WORK( LDWORK, NBMAX ) 00173 * .. 00174 * .. External Functions .. 00175 LOGICAL LSAME 00176 INTEGER ILAENV 00177 EXTERNAL LSAME, ILAENV 00178 * .. 00179 * .. External Subroutines .. 00180 EXTERNAL CGEMM, CHERK, CPBTF2, CPOTF2, CTRSM, XERBLA 00181 * .. 00182 * .. Intrinsic Functions .. 00183 INTRINSIC MIN 00184 * .. 00185 * .. Executable Statements .. 00186 * 00187 * Test the input parameters. 00188 * 00189 INFO = 0 00190 IF( ( .NOT.LSAME( UPLO, 'U' ) ) .AND. 00191 $ ( .NOT.LSAME( UPLO, 'L' ) ) ) THEN 00192 INFO = -1 00193 ELSE IF( N.LT.0 ) THEN 00194 INFO = -2 00195 ELSE IF( KD.LT.0 ) THEN 00196 INFO = -3 00197 ELSE IF( LDAB.LT.KD+1 ) THEN 00198 INFO = -5 00199 END IF 00200 IF( INFO.NE.0 ) THEN 00201 CALL XERBLA( 'CPBTRF', -INFO ) 00202 RETURN 00203 END IF 00204 * 00205 * Quick return if possible 00206 * 00207 IF( N.EQ.0 ) 00208 $ RETURN 00209 * 00210 * Determine the block size for this environment 00211 * 00212 NB = ILAENV( 1, 'CPBTRF', UPLO, N, KD, -1, -1 ) 00213 * 00214 * The block size must not exceed the semi-bandwidth KD, and must not 00215 * exceed the limit set by the size of the local array WORK. 00216 * 00217 NB = MIN( NB, NBMAX ) 00218 * 00219 IF( NB.LE.1 .OR. NB.GT.KD ) THEN 00220 * 00221 * Use unblocked code 00222 * 00223 CALL CPBTF2( UPLO, N, KD, AB, LDAB, INFO ) 00224 ELSE 00225 * 00226 * Use blocked code 00227 * 00228 IF( LSAME( UPLO, 'U' ) ) THEN 00229 * 00230 * Compute the Cholesky factorization of a Hermitian band 00231 * matrix, given the upper triangle of the matrix in band 00232 * storage. 00233 * 00234 * Zero the upper triangle of the work array. 00235 * 00236 DO 20 J = 1, NB 00237 DO 10 I = 1, J - 1 00238 WORK( I, J ) = ZERO 00239 10 CONTINUE 00240 20 CONTINUE 00241 * 00242 * Process the band matrix one diagonal block at a time. 00243 * 00244 DO 70 I = 1, N, NB 00245 IB = MIN( NB, N-I+1 ) 00246 * 00247 * Factorize the diagonal block 00248 * 00249 CALL CPOTF2( UPLO, IB, AB( KD+1, I ), LDAB-1, II ) 00250 IF( II.NE.0 ) THEN 00251 INFO = I + II - 1 00252 GO TO 150 00253 END IF 00254 IF( I+IB.LE.N ) THEN 00255 * 00256 * Update the relevant part of the trailing submatrix. 00257 * If A11 denotes the diagonal block which has just been 00258 * factorized, then we need to update the remaining 00259 * blocks in the diagram: 00260 * 00261 * A11 A12 A13 00262 * A22 A23 00263 * A33 00264 * 00265 * The numbers of rows and columns in the partitioning 00266 * are IB, I2, I3 respectively. The blocks A12, A22 and 00267 * A23 are empty if IB = KD. The upper triangle of A13 00268 * lies outside the band. 00269 * 00270 I2 = MIN( KD-IB, N-I-IB+1 ) 00271 I3 = MIN( IB, N-I-KD+1 ) 00272 * 00273 IF( I2.GT.0 ) THEN 00274 * 00275 * Update A12 00276 * 00277 CALL CTRSM( 'Left', 'Upper', 'Conjugate transpose', 00278 $ 'Non-unit', IB, I2, CONE, 00279 $ AB( KD+1, I ), LDAB-1, 00280 $ AB( KD+1-IB, I+IB ), LDAB-1 ) 00281 * 00282 * Update A22 00283 * 00284 CALL CHERK( 'Upper', 'Conjugate transpose', I2, IB, 00285 $ -ONE, AB( KD+1-IB, I+IB ), LDAB-1, ONE, 00286 $ AB( KD+1, I+IB ), LDAB-1 ) 00287 END IF 00288 * 00289 IF( I3.GT.0 ) THEN 00290 * 00291 * Copy the lower triangle of A13 into the work array. 00292 * 00293 DO 40 JJ = 1, I3 00294 DO 30 II = JJ, IB 00295 WORK( II, JJ ) = AB( II-JJ+1, JJ+I+KD-1 ) 00296 30 CONTINUE 00297 40 CONTINUE 00298 * 00299 * Update A13 (in the work array). 00300 * 00301 CALL CTRSM( 'Left', 'Upper', 'Conjugate transpose', 00302 $ 'Non-unit', IB, I3, CONE, 00303 $ AB( KD+1, I ), LDAB-1, WORK, LDWORK ) 00304 * 00305 * Update A23 00306 * 00307 IF( I2.GT.0 ) 00308 $ CALL CGEMM( 'Conjugate transpose', 00309 $ 'No transpose', I2, I3, IB, -CONE, 00310 $ AB( KD+1-IB, I+IB ), LDAB-1, WORK, 00311 $ LDWORK, CONE, AB( 1+IB, I+KD ), 00312 $ LDAB-1 ) 00313 * 00314 * Update A33 00315 * 00316 CALL CHERK( 'Upper', 'Conjugate transpose', I3, IB, 00317 $ -ONE, WORK, LDWORK, ONE, 00318 $ AB( KD+1, I+KD ), LDAB-1 ) 00319 * 00320 * Copy the lower triangle of A13 back into place. 00321 * 00322 DO 60 JJ = 1, I3 00323 DO 50 II = JJ, IB 00324 AB( II-JJ+1, JJ+I+KD-1 ) = WORK( II, JJ ) 00325 50 CONTINUE 00326 60 CONTINUE 00327 END IF 00328 END IF 00329 70 CONTINUE 00330 ELSE 00331 * 00332 * Compute the Cholesky factorization of a Hermitian band 00333 * matrix, given the lower triangle of the matrix in band 00334 * storage. 00335 * 00336 * Zero the lower triangle of the work array. 00337 * 00338 DO 90 J = 1, NB 00339 DO 80 I = J + 1, NB 00340 WORK( I, J ) = ZERO 00341 80 CONTINUE 00342 90 CONTINUE 00343 * 00344 * Process the band matrix one diagonal block at a time. 00345 * 00346 DO 140 I = 1, N, NB 00347 IB = MIN( NB, N-I+1 ) 00348 * 00349 * Factorize the diagonal block 00350 * 00351 CALL CPOTF2( UPLO, IB, AB( 1, I ), LDAB-1, II ) 00352 IF( II.NE.0 ) THEN 00353 INFO = I + II - 1 00354 GO TO 150 00355 END IF 00356 IF( I+IB.LE.N ) THEN 00357 * 00358 * Update the relevant part of the trailing submatrix. 00359 * If A11 denotes the diagonal block which has just been 00360 * factorized, then we need to update the remaining 00361 * blocks in the diagram: 00362 * 00363 * A11 00364 * A21 A22 00365 * A31 A32 A33 00366 * 00367 * The numbers of rows and columns in the partitioning 00368 * are IB, I2, I3 respectively. The blocks A21, A22 and 00369 * A32 are empty if IB = KD. The lower triangle of A31 00370 * lies outside the band. 00371 * 00372 I2 = MIN( KD-IB, N-I-IB+1 ) 00373 I3 = MIN( IB, N-I-KD+1 ) 00374 * 00375 IF( I2.GT.0 ) THEN 00376 * 00377 * Update A21 00378 * 00379 CALL CTRSM( 'Right', 'Lower', 00380 $ 'Conjugate transpose', 'Non-unit', I2, 00381 $ IB, CONE, AB( 1, I ), LDAB-1, 00382 $ AB( 1+IB, I ), LDAB-1 ) 00383 * 00384 * Update A22 00385 * 00386 CALL CHERK( 'Lower', 'No transpose', I2, IB, -ONE, 00387 $ AB( 1+IB, I ), LDAB-1, ONE, 00388 $ AB( 1, I+IB ), LDAB-1 ) 00389 END IF 00390 * 00391 IF( I3.GT.0 ) THEN 00392 * 00393 * Copy the upper triangle of A31 into the work array. 00394 * 00395 DO 110 JJ = 1, IB 00396 DO 100 II = 1, MIN( JJ, I3 ) 00397 WORK( II, JJ ) = AB( KD+1-JJ+II, JJ+I-1 ) 00398 100 CONTINUE 00399 110 CONTINUE 00400 * 00401 * Update A31 (in the work array). 00402 * 00403 CALL CTRSM( 'Right', 'Lower', 00404 $ 'Conjugate transpose', 'Non-unit', I3, 00405 $ IB, CONE, AB( 1, I ), LDAB-1, WORK, 00406 $ LDWORK ) 00407 * 00408 * Update A32 00409 * 00410 IF( I2.GT.0 ) 00411 $ CALL CGEMM( 'No transpose', 00412 $ 'Conjugate transpose', I3, I2, IB, 00413 $ -CONE, WORK, LDWORK, AB( 1+IB, I ), 00414 $ LDAB-1, CONE, AB( 1+KD-IB, I+IB ), 00415 $ LDAB-1 ) 00416 * 00417 * Update A33 00418 * 00419 CALL CHERK( 'Lower', 'No transpose', I3, IB, -ONE, 00420 $ WORK, LDWORK, ONE, AB( 1, I+KD ), 00421 $ LDAB-1 ) 00422 * 00423 * Copy the upper triangle of A31 back into place. 00424 * 00425 DO 130 JJ = 1, IB 00426 DO 120 II = 1, MIN( JJ, I3 ) 00427 AB( KD+1-JJ+II, JJ+I-1 ) = WORK( II, JJ ) 00428 120 CONTINUE 00429 130 CONTINUE 00430 END IF 00431 END IF 00432 140 CONTINUE 00433 END IF 00434 END IF 00435 RETURN 00436 * 00437 150 CONTINUE 00438 RETURN 00439 * 00440 * End of CPBTRF 00441 * 00442 END