1 SUBROUTINE CPBTRF( UPLO, N, KD, AB, LDAB, INFO ) 2* 3* -- LAPACK routine (version 3.0) -- 4* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., 5* Courant Institute, Argonne National Lab, and Rice University 6* September 30, 1994 7* 8* .. Scalar Arguments .. 9 CHARACTER UPLO 10 INTEGER INFO, KD, LDAB, N 11* .. 12* .. Array Arguments .. 13 COMPLEX AB( LDAB, * ) 14* .. 15* 16* Purpose 17* ======= 18* 19* CPBTRF computes the Cholesky factorization of a complex Hermitian 20* positive definite band matrix A. 21* 22* The factorization has the form 23* A = U**H * U, if UPLO = 'U', or 24* A = L * L**H, if UPLO = 'L', 25* where U is an upper triangular matrix and L is lower triangular. 26* 27* Arguments 28* ========= 29* 30* UPLO (input) CHARACTER*1 31* = 'U': Upper triangle of A is stored; 32* = 'L': Lower triangle of A is stored. 33* 34* N (input) INTEGER 35* The order of the matrix A. N >= 0. 36* 37* KD (input) INTEGER 38* The number of superdiagonals of the matrix A if UPLO = 'U', 39* or the number of subdiagonals if UPLO = 'L'. KD >= 0. 40* 41* AB (input/output) COMPLEX array, dimension (LDAB,N) 42* On entry, the upper or lower triangle of the Hermitian band 43* matrix A, stored in the first KD+1 rows of the array. The 44* j-th column of A is stored in the j-th column of the array AB 45* as follows: 46* if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; 47* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). 48* 49* On exit, if INFO = 0, the triangular factor U or L from the 50* Cholesky factorization A = U**H*U or A = L*L**H of the band 51* matrix A, in the same storage format as A. 52* 53* LDAB (input) INTEGER 54* The leading dimension of the array AB. LDAB >= KD+1. 55* 56* INFO (output) INTEGER 57* = 0: successful exit 58* < 0: if INFO = -i, the i-th argument had an illegal value 59* > 0: if INFO = i, the leading minor of order i is not 60* positive definite, and the factorization could not be 61* completed. 62* 63* Further Details 64* =============== 65* 66* The band storage scheme is illustrated by the following example, when 67* N = 6, KD = 2, and UPLO = 'U': 68* 69* On entry: On exit: 70* 71* * * a13 a24 a35 a46 * * u13 u24 u35 u46 72* * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 73* a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 74* 75* Similarly, if UPLO = 'L' the format of A is as follows: 76* 77* On entry: On exit: 78* 79* a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66 80* a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 * 81* a31 a42 a53 a64 * * l31 l42 l53 l64 * * 82* 83* Array elements marked * are not used by the routine. 84* 85* Contributed by 86* Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989 87* 88* ===================================================================== 89* 90* .. Parameters .. 91 REAL ONE, ZERO 92 PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 ) 93 COMPLEX CONE 94 PARAMETER ( CONE = ( 1.0E+0, 0.0E+0 ) ) 95 INTEGER NBMAX, LDWORK 96 PARAMETER ( NBMAX = 32, LDWORK = NBMAX+1 ) 97* .. 98* .. Local Scalars .. 99 INTEGER I, I2, I3, IB, II, J, JJ, NB 100* .. 101* .. Local Arrays .. 102 COMPLEX WORK( LDWORK, NBMAX ) 103* .. 104* .. External Functions .. 105 LOGICAL LSAME 106 INTEGER ILAENV 107 EXTERNAL LSAME, ILAENV 108* .. 109* .. External Subroutines .. 110 EXTERNAL CGEMM, CHERK, CPBTF2, CPOTF2, CTRSM, XERBLA 111* .. 112* .. Intrinsic Functions .. 113 INTRINSIC MIN 114* .. 115* .. Executable Statements .. 116* 117* Test the input parameters. 118* 119 INFO = 0 120 IF( ( .NOT.LSAME( UPLO, 'U' ) ) .AND. 121 $ ( .NOT.LSAME( UPLO, 'L' ) ) ) THEN 122 INFO = -1 123 ELSE IF( N.LT.0 ) THEN 124 INFO = -2 125 ELSE IF( KD.LT.0 ) THEN 126 INFO = -3 127 ELSE IF( LDAB.LT.KD+1 ) THEN 128 INFO = -5 129 END IF 130 IF( INFO.NE.0 ) THEN 131 CALL XERBLA( 'CPBTRF', -INFO ) 132 RETURN 133 END IF 134* 135* Quick return if possible 136* 137 IF( N.EQ.0 ) 138 $ RETURN 139* 140* Determine the block size for this environment 141* 142 NB = ILAENV( 1, 'CPBTRF', UPLO, N, KD, -1, -1 ) 143* 144* The block size must not exceed the semi-bandwidth KD, and must not 145* exceed the limit set by the size of the local array WORK. 146* 147 NB = MIN( NB, NBMAX ) 148* 149 IF( NB.LE.1 .OR. NB.GT.KD ) THEN 150* 151* Use unblocked code 152* 153 CALL CPBTF2( UPLO, N, KD, AB, LDAB, INFO ) 154 ELSE 155* 156* Use blocked code 157* 158 IF( LSAME( UPLO, 'U' ) ) THEN 159* 160* Compute the Cholesky factorization of a Hermitian band 161* matrix, given the upper triangle of the matrix in band 162* storage. 163* 164* Zero the upper triangle of the work array. 165* 166 DO 20 J = 1, NB 167 DO 10 I = 1, J - 1 168 WORK( I, J ) = ZERO 169 10 CONTINUE 170 20 CONTINUE 171* 172* Process the band matrix one diagonal block at a time. 173* 174 DO 70 I = 1, N, NB 175 IB = MIN( NB, N-I+1 ) 176* 177* Factorize the diagonal block 178* 179 CALL CPOTF2( UPLO, IB, AB( KD+1, I ), LDAB-1, II ) 180 IF( II.NE.0 ) THEN 181 INFO = I + II - 1 182 GO TO 150 183 END IF 184 IF( I+IB.LE.N ) THEN 185* 186* Update the relevant part of the trailing submatrix. 187* If A11 denotes the diagonal block which has just been 188* factorized, then we need to update the remaining 189* blocks in the diagram: 190* 191* A11 A12 A13 192* A22 A23 193* A33 194* 195* The numbers of rows and columns in the partitioning 196* are IB, I2, I3 respectively. The blocks A12, A22 and 197* A23 are empty if IB = KD. The upper triangle of A13 198* lies outside the band. 199* 200 I2 = MIN( KD-IB, N-I-IB+1 ) 201 I3 = MIN( IB, N-I-KD+1 ) 202* 203 IF( I2.GT.0 ) THEN 204* 205* Update A12 206* 207 CALL CTRSM( 'Left', 'Upper', 'Conjugate transpose', 208 $ 'Non-unit', IB, I2, CONE, 209 $ AB( KD+1, I ), LDAB-1, 210 $ AB( KD+1-IB, I+IB ), LDAB-1 ) 211* 212* Update A22 213* 214 CALL CHERK( 'Upper', 'Conjugate transpose', I2, IB, 215 $ -ONE, AB( KD+1-IB, I+IB ), LDAB-1, ONE, 216 $ AB( KD+1, I+IB ), LDAB-1 ) 217 END IF 218* 219 IF( I3.GT.0 ) THEN 220* 221* Copy the lower triangle of A13 into the work array. 222* 223 DO 40 JJ = 1, I3 224 DO 30 II = JJ, IB 225 WORK( II, JJ ) = AB( II-JJ+1, JJ+I+KD-1 ) 226 30 CONTINUE 227 40 CONTINUE 228* 229* Update A13 (in the work array). 230* 231 CALL CTRSM( 'Left', 'Upper', 'Conjugate transpose', 232 $ 'Non-unit', IB, I3, CONE, 233 $ AB( KD+1, I ), LDAB-1, WORK, LDWORK ) 234* 235* Update A23 236* 237 IF( I2.GT.0 ) 238 $ CALL CGEMM( 'Conjugate transpose', 239 $ 'No transpose', I2, I3, IB, -CONE, 240 $ AB( KD+1-IB, I+IB ), LDAB-1, WORK, 241 $ LDWORK, CONE, AB( 1+IB, I+KD ), 242 $ LDAB-1 ) 243* 244* Update A33 245* 246 CALL CHERK( 'Upper', 'Conjugate transpose', I3, IB, 247 $ -ONE, WORK, LDWORK, ONE, 248 $ AB( KD+1, I+KD ), LDAB-1 ) 249* 250* Copy the lower triangle of A13 back into place. 251* 252 DO 60 JJ = 1, I3 253 DO 50 II = JJ, IB 254 AB( II-JJ+1, JJ+I+KD-1 ) = WORK( II, JJ ) 255 50 CONTINUE 256 60 CONTINUE 257 END IF 258 END IF 259 70 CONTINUE 260 ELSE 261* 262* Compute the Cholesky factorization of a Hermitian band 263* matrix, given the lower triangle of the matrix in band 264* storage. 265* 266* Zero the lower triangle of the work array. 267* 268 DO 90 J = 1, NB 269 DO 80 I = J + 1, NB 270 WORK( I, J ) = ZERO 271 80 CONTINUE 272 90 CONTINUE 273* 274* Process the band matrix one diagonal block at a time. 275* 276 DO 140 I = 1, N, NB 277 IB = MIN( NB, N-I+1 ) 278* 279* Factorize the diagonal block 280* 281 CALL CPOTF2( UPLO, IB, AB( 1, I ), LDAB-1, II ) 282 IF( II.NE.0 ) THEN 283 INFO = I + II - 1 284 GO TO 150 285 END IF 286 IF( I+IB.LE.N ) THEN 287* 288* Update the relevant part of the trailing submatrix. 289* If A11 denotes the diagonal block which has just been 290* factorized, then we need to update the remaining 291* blocks in the diagram: 292* 293* A11 294* A21 A22 295* A31 A32 A33 296* 297* The numbers of rows and columns in the partitioning 298* are IB, I2, I3 respectively. The blocks A21, A22 and 299* A32 are empty if IB = KD. The lower triangle of A31 300* lies outside the band. 301* 302 I2 = MIN( KD-IB, N-I-IB+1 ) 303 I3 = MIN( IB, N-I-KD+1 ) 304* 305 IF( I2.GT.0 ) THEN 306* 307* Update A21 308* 309 CALL CTRSM( 'Right', 'Lower', 310 $ 'Conjugate transpose', 'Non-unit', I2, 311 $ IB, CONE, AB( 1, I ), LDAB-1, 312 $ AB( 1+IB, I ), LDAB-1 ) 313* 314* Update A22 315* 316 CALL CHERK( 'Lower', 'No transpose', I2, IB, -ONE, 317 $ AB( 1+IB, I ), LDAB-1, ONE, 318 $ AB( 1, I+IB ), LDAB-1 ) 319 END IF 320* 321 IF( I3.GT.0 ) THEN 322* 323* Copy the upper triangle of A31 into the work array. 324* 325 DO 110 JJ = 1, IB 326 DO 100 II = 1, MIN( JJ, I3 ) 327 WORK( II, JJ ) = AB( KD+1-JJ+II, JJ+I-1 ) 328 100 CONTINUE 329 110 CONTINUE 330* 331* Update A31 (in the work array). 332* 333 CALL CTRSM( 'Right', 'Lower', 334 $ 'Conjugate transpose', 'Non-unit', I3, 335 $ IB, CONE, AB( 1, I ), LDAB-1, WORK, 336 $ LDWORK ) 337* 338* Update A32 339* 340 IF( I2.GT.0 ) 341 $ CALL CGEMM( 'No transpose', 342 $ 'Conjugate transpose', I3, I2, IB, 343 $ -CONE, WORK, LDWORK, AB( 1+IB, I ), 344 $ LDAB-1, CONE, AB( 1+KD-IB, I+IB ), 345 $ LDAB-1 ) 346* 347* Update A33 348* 349 CALL CHERK( 'Lower', 'No transpose', I3, IB, -ONE, 350 $ WORK, LDWORK, ONE, AB( 1, I+KD ), 351 $ LDAB-1 ) 352* 353* Copy the upper triangle of A31 back into place. 354* 355 DO 130 JJ = 1, IB 356 DO 120 II = 1, MIN( JJ, I3 ) 357 AB( KD+1-JJ+II, JJ+I-1 ) = WORK( II, JJ ) 358 120 CONTINUE 359 130 CONTINUE 360 END IF 361 END IF 362 140 CONTINUE 363 END IF 364 END IF 365 RETURN 366* 367 150 CONTINUE 368 RETURN 369* 370* End of CPBTRF 371* 372 END 373