1 SUBROUTINE SPBTRF( 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* March 31, 1993 7* 8* .. Scalar Arguments .. 9 CHARACTER UPLO 10 INTEGER INFO, KD, LDAB, N 11* .. 12* .. Array Arguments .. 13 REAL AB( LDAB, * ) 14* .. 15* 16* Purpose 17* ======= 18* 19* SPBTRF computes the Cholesky factorization of a real symmetric 20* positive definite band matrix A. 21* 22* The factorization has the form 23* A = U**T * U, if UPLO = 'U', or 24* A = L * L**T, 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) REAL array, dimension (LDAB,N) 42* On entry, the upper or lower triangle of the symmetric 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**T*U or A = L*L**T 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 INTEGER NBMAX, LDWORK 94 PARAMETER ( NBMAX = 32, LDWORK = NBMAX+1 ) 95* .. 96* .. Local Scalars .. 97 INTEGER I, I2, I3, IB, II, J, JJ, NB 98* .. 99* .. Local Arrays .. 100 REAL WORK( LDWORK, NBMAX ) 101* .. 102* .. External Functions .. 103 LOGICAL LSAME 104 INTEGER ILAENV 105 EXTERNAL LSAME, ILAENV 106* .. 107* .. External Subroutines .. 108 EXTERNAL SGEMM, SPBTF2, SPOTF2, SSYRK, STRSM, XERBLA 109* .. 110* .. Intrinsic Functions .. 111 INTRINSIC MIN 112* .. 113* .. Executable Statements .. 114* 115* Test the input parameters. 116* 117 INFO = 0 118 IF( ( .NOT.LSAME( UPLO, 'U' ) ) .AND. 119 $ ( .NOT.LSAME( UPLO, 'L' ) ) ) THEN 120 INFO = -1 121 ELSE IF( N.LT.0 ) THEN 122 INFO = -2 123 ELSE IF( KD.LT.0 ) THEN 124 INFO = -3 125 ELSE IF( LDAB.LT.KD+1 ) THEN 126 INFO = -5 127 END IF 128 IF( INFO.NE.0 ) THEN 129 CALL XERBLA( 'SPBTRF', -INFO ) 130 RETURN 131 END IF 132* 133* Quick return if possible 134* 135 IF( N.EQ.0 ) 136 $ RETURN 137* 138* Determine the block size for this environment 139* 140 NB = ILAENV( 1, 'SPBTRF', UPLO, N, KD, -1, -1 ) 141* 142* The block size must not exceed the semi-bandwidth KD, and must not 143* exceed the limit set by the size of the local array WORK. 144* 145 NB = MIN( NB, NBMAX ) 146* 147 IF( NB.LE.1 .OR. NB.GT.KD ) THEN 148* 149* Use unblocked code 150* 151 CALL SPBTF2( UPLO, N, KD, AB, LDAB, INFO ) 152 ELSE 153* 154* Use blocked code 155* 156 IF( LSAME( UPLO, 'U' ) ) THEN 157* 158* Compute the Cholesky factorization of a symmetric band 159* matrix, given the upper triangle of the matrix in band 160* storage. 161* 162* Zero the upper triangle of the work array. 163* 164 DO 20 J = 1, NB 165 DO 10 I = 1, J - 1 166 WORK( I, J ) = ZERO 167 10 CONTINUE 168 20 CONTINUE 169* 170* Process the band matrix one diagonal block at a time. 171* 172 DO 70 I = 1, N, NB 173 IB = MIN( NB, N-I+1 ) 174* 175* Factorize the diagonal block 176* 177 CALL SPOTF2( UPLO, IB, AB( KD+1, I ), LDAB-1, II ) 178 IF( II.NE.0 ) THEN 179 INFO = I + II - 1 180 GO TO 150 181 END IF 182 IF( I+IB.LE.N ) THEN 183* 184* Update the relevant part of the trailing submatrix. 185* If A11 denotes the diagonal block which has just been 186* factorized, then we need to update the remaining 187* blocks in the diagram: 188* 189* A11 A12 A13 190* A22 A23 191* A33 192* 193* The numbers of rows and columns in the partitioning 194* are IB, I2, I3 respectively. The blocks A12, A22 and 195* A23 are empty if IB = KD. The upper triangle of A13 196* lies outside the band. 197* 198 I2 = MIN( KD-IB, N-I-IB+1 ) 199 I3 = MIN( IB, N-I-KD+1 ) 200* 201 IF( I2.GT.0 ) THEN 202* 203* Update A12 204* 205 CALL STRSM( 'Left', 'Upper', 'Transpose', 206 $ 'Non-unit', IB, I2, ONE, AB( KD+1, I ), 207 $ LDAB-1, AB( KD+1-IB, I+IB ), LDAB-1 ) 208* 209* Update A22 210* 211 CALL SSYRK( 'Upper', 'Transpose', I2, IB, -ONE, 212 $ AB( KD+1-IB, I+IB ), LDAB-1, ONE, 213 $ AB( KD+1, I+IB ), LDAB-1 ) 214 END IF 215* 216 IF( I3.GT.0 ) THEN 217* 218* Copy the lower triangle of A13 into the work array. 219* 220 DO 40 JJ = 1, I3 221 DO 30 II = JJ, IB 222 WORK( II, JJ ) = AB( II-JJ+1, JJ+I+KD-1 ) 223 30 CONTINUE 224 40 CONTINUE 225* 226* Update A13 (in the work array). 227* 228 CALL STRSM( 'Left', 'Upper', 'Transpose', 229 $ 'Non-unit', IB, I3, ONE, AB( KD+1, I ), 230 $ LDAB-1, WORK, LDWORK ) 231* 232* Update A23 233* 234 IF( I2.GT.0 ) 235 $ CALL SGEMM( 'Transpose', 'No Transpose', I2, I3, 236 $ IB, -ONE, AB( KD+1-IB, I+IB ), 237 $ LDAB-1, WORK, LDWORK, ONE, 238 $ AB( 1+IB, I+KD ), LDAB-1 ) 239* 240* Update A33 241* 242 CALL SSYRK( 'Upper', 'Transpose', I3, IB, -ONE, 243 $ WORK, LDWORK, ONE, AB( KD+1, I+KD ), 244 $ LDAB-1 ) 245* 246* Copy the lower triangle of A13 back into place. 247* 248 DO 60 JJ = 1, I3 249 DO 50 II = JJ, IB 250 AB( II-JJ+1, JJ+I+KD-1 ) = WORK( II, JJ ) 251 50 CONTINUE 252 60 CONTINUE 253 END IF 254 END IF 255 70 CONTINUE 256 ELSE 257* 258* Compute the Cholesky factorization of a symmetric band 259* matrix, given the lower triangle of the matrix in band 260* storage. 261* 262* Zero the lower triangle of the work array. 263* 264 DO 90 J = 1, NB 265 DO 80 I = J + 1, NB 266 WORK( I, J ) = ZERO 267 80 CONTINUE 268 90 CONTINUE 269* 270* Process the band matrix one diagonal block at a time. 271* 272 DO 140 I = 1, N, NB 273 IB = MIN( NB, N-I+1 ) 274* 275* Factorize the diagonal block 276* 277 CALL SPOTF2( UPLO, IB, AB( 1, I ), LDAB-1, II ) 278 IF( II.NE.0 ) THEN 279 INFO = I + II - 1 280 GO TO 150 281 END IF 282 IF( I+IB.LE.N ) THEN 283* 284* Update the relevant part of the trailing submatrix. 285* If A11 denotes the diagonal block which has just been 286* factorized, then we need to update the remaining 287* blocks in the diagram: 288* 289* A11 290* A21 A22 291* A31 A32 A33 292* 293* The numbers of rows and columns in the partitioning 294* are IB, I2, I3 respectively. The blocks A21, A22 and 295* A32 are empty if IB = KD. The lower triangle of A31 296* lies outside the band. 297* 298 I2 = MIN( KD-IB, N-I-IB+1 ) 299 I3 = MIN( IB, N-I-KD+1 ) 300* 301 IF( I2.GT.0 ) THEN 302* 303* Update A21 304* 305 CALL STRSM( 'Right', 'Lower', 'Transpose', 306 $ 'Non-unit', I2, IB, ONE, AB( 1, I ), 307 $ LDAB-1, AB( 1+IB, I ), LDAB-1 ) 308* 309* Update A22 310* 311 CALL SSYRK( 'Lower', 'No Transpose', I2, IB, -ONE, 312 $ AB( 1+IB, I ), LDAB-1, ONE, 313 $ AB( 1, I+IB ), LDAB-1 ) 314 END IF 315* 316 IF( I3.GT.0 ) THEN 317* 318* Copy the upper triangle of A31 into the work array. 319* 320 DO 110 JJ = 1, IB 321 DO 100 II = 1, MIN( JJ, I3 ) 322 WORK( II, JJ ) = AB( KD+1-JJ+II, JJ+I-1 ) 323 100 CONTINUE 324 110 CONTINUE 325* 326* Update A31 (in the work array). 327* 328 CALL STRSM( 'Right', 'Lower', 'Transpose', 329 $ 'Non-unit', I3, IB, ONE, AB( 1, I ), 330 $ LDAB-1, WORK, LDWORK ) 331* 332* Update A32 333* 334 IF( I2.GT.0 ) 335 $ CALL SGEMM( 'No transpose', 'Transpose', I3, I2, 336 $ IB, -ONE, WORK, LDWORK, 337 $ AB( 1+IB, I ), LDAB-1, ONE, 338 $ AB( 1+KD-IB, I+IB ), LDAB-1 ) 339* 340* Update A33 341* 342 CALL SSYRK( 'Lower', 'No Transpose', I3, IB, -ONE, 343 $ WORK, LDWORK, ONE, AB( 1, I+KD ), 344 $ LDAB-1 ) 345* 346* Copy the upper triangle of A31 back into place. 347* 348 DO 130 JJ = 1, IB 349 DO 120 II = 1, MIN( JJ, I3 ) 350 AB( KD+1-JJ+II, JJ+I-1 ) = WORK( II, JJ ) 351 120 CONTINUE 352 130 CONTINUE 353 END IF 354 END IF 355 140 CONTINUE 356 END IF 357 END IF 358 RETURN 359* 360 150 CONTINUE 361 RETURN 362* 363* End of SPBTRF 364* 365 END 366