1*> \brief \b SPBSTF 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8*> \htmlonly 9*> Download SPBSTF + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/spbstf.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/spbstf.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/spbstf.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE SPBSTF( UPLO, N, KD, AB, LDAB, INFO ) 22* 23* .. Scalar Arguments .. 24* CHARACTER UPLO 25* INTEGER INFO, KD, LDAB, N 26* .. 27* .. Array Arguments .. 28* REAL AB( LDAB, * ) 29* .. 30* 31* 32*> \par Purpose: 33* ============= 34*> 35*> \verbatim 36*> 37*> SPBSTF computes a split Cholesky factorization of a real 38*> symmetric positive definite band matrix A. 39*> 40*> This routine is designed to be used in conjunction with SSBGST. 41*> 42*> The factorization has the form A = S**T*S where S is a band matrix 43*> of the same bandwidth as A and the following structure: 44*> 45*> S = ( U ) 46*> ( M L ) 47*> 48*> where U is upper triangular of order m = (n+kd)/2, and L is lower 49*> triangular of order n-m. 50*> \endverbatim 51* 52* Arguments: 53* ========== 54* 55*> \param[in] UPLO 56*> \verbatim 57*> UPLO is CHARACTER*1 58*> = 'U': Upper triangle of A is stored; 59*> = 'L': Lower triangle of A is stored. 60*> \endverbatim 61*> 62*> \param[in] N 63*> \verbatim 64*> N is INTEGER 65*> The order of the matrix A. N >= 0. 66*> \endverbatim 67*> 68*> \param[in] KD 69*> \verbatim 70*> KD is INTEGER 71*> The number of superdiagonals of the matrix A if UPLO = 'U', 72*> or the number of subdiagonals if UPLO = 'L'. KD >= 0. 73*> \endverbatim 74*> 75*> \param[in,out] AB 76*> \verbatim 77*> AB is REAL array, dimension (LDAB,N) 78*> On entry, the upper or lower triangle of the symmetric band 79*> matrix A, stored in the first kd+1 rows of the array. The 80*> j-th column of A is stored in the j-th column of the array AB 81*> as follows: 82*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; 83*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). 84*> 85*> On exit, if INFO = 0, the factor S from the split Cholesky 86*> factorization A = S**T*S. See Further Details. 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 = -i, the i-th argument had an illegal value 100*> > 0: if INFO = i, the factorization could not be completed, 101*> because the updated element a(i,i) was negative; the 102*> matrix A is not positive definite. 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 realOTHERcomputational 114* 115*> \par Further Details: 116* ===================== 117*> 118*> \verbatim 119*> 120*> The band storage scheme is illustrated by the following example, when 121*> N = 7, KD = 2: 122*> 123*> S = ( s11 s12 s13 ) 124*> ( s22 s23 s24 ) 125*> ( s33 s34 ) 126*> ( s44 ) 127*> ( s53 s54 s55 ) 128*> ( s64 s65 s66 ) 129*> ( s75 s76 s77 ) 130*> 131*> If UPLO = 'U', the array AB holds: 132*> 133*> on entry: on exit: 134*> 135*> * * a13 a24 a35 a46 a57 * * s13 s24 s53 s64 s75 136*> * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54 s65 s76 137*> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 138*> 139*> If UPLO = 'L', the array AB holds: 140*> 141*> on entry: on exit: 142*> 143*> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 144*> a21 a32 a43 a54 a65 a76 * s12 s23 s34 s54 s65 s76 * 145*> a31 a42 a53 a64 a64 * * s13 s24 s53 s64 s75 * * 146*> 147*> Array elements marked * are not used by the routine. 148*> \endverbatim 149*> 150* ===================================================================== 151 SUBROUTINE SPBSTF( UPLO, N, KD, AB, LDAB, INFO ) 152* 153* -- LAPACK computational routine -- 154* -- LAPACK is a software package provided by Univ. of Tennessee, -- 155* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 156* 157* .. Scalar Arguments .. 158 CHARACTER UPLO 159 INTEGER INFO, KD, LDAB, N 160* .. 161* .. Array Arguments .. 162 REAL AB( LDAB, * ) 163* .. 164* 165* ===================================================================== 166* 167* .. Parameters .. 168 REAL ONE, ZERO 169 PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 ) 170* .. 171* .. Local Scalars .. 172 LOGICAL UPPER 173 INTEGER J, KLD, KM, M 174 REAL AJJ 175* .. 176* .. External Functions .. 177 LOGICAL LSAME 178 EXTERNAL LSAME 179* .. 180* .. External Subroutines .. 181 EXTERNAL SSCAL, SSYR, XERBLA 182* .. 183* .. Intrinsic Functions .. 184 INTRINSIC MAX, MIN, SQRT 185* .. 186* .. Executable Statements .. 187* 188* Test the input parameters. 189* 190 INFO = 0 191 UPPER = LSAME( UPLO, 'U' ) 192 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 193 INFO = -1 194 ELSE IF( N.LT.0 ) THEN 195 INFO = -2 196 ELSE IF( KD.LT.0 ) THEN 197 INFO = -3 198 ELSE IF( LDAB.LT.KD+1 ) THEN 199 INFO = -5 200 END IF 201 IF( INFO.NE.0 ) THEN 202 CALL XERBLA( 'SPBSTF', -INFO ) 203 RETURN 204 END IF 205* 206* Quick return if possible 207* 208 IF( N.EQ.0 ) 209 $ RETURN 210* 211 KLD = MAX( 1, LDAB-1 ) 212* 213* Set the splitting point m. 214* 215 M = ( N+KD ) / 2 216* 217 IF( UPPER ) THEN 218* 219* Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m). 220* 221 DO 10 J = N, M + 1, -1 222* 223* Compute s(j,j) and test for non-positive-definiteness. 224* 225 AJJ = AB( KD+1, J ) 226 IF( AJJ.LE.ZERO ) 227 $ GO TO 50 228 AJJ = SQRT( AJJ ) 229 AB( KD+1, J ) = AJJ 230 KM = MIN( J-1, KD ) 231* 232* Compute elements j-km:j-1 of the j-th column and update the 233* the leading submatrix within the band. 234* 235 CALL SSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 ) 236 CALL SSYR( 'Upper', KM, -ONE, AB( KD+1-KM, J ), 1, 237 $ AB( KD+1, J-KM ), KLD ) 238 10 CONTINUE 239* 240* Factorize the updated submatrix A(1:m,1:m) as U**T*U. 241* 242 DO 20 J = 1, M 243* 244* Compute s(j,j) and test for non-positive-definiteness. 245* 246 AJJ = AB( KD+1, J ) 247 IF( AJJ.LE.ZERO ) 248 $ GO TO 50 249 AJJ = SQRT( AJJ ) 250 AB( KD+1, J ) = AJJ 251 KM = MIN( KD, M-J ) 252* 253* Compute elements j+1:j+km of the j-th row and update the 254* trailing submatrix within the band. 255* 256 IF( KM.GT.0 ) THEN 257 CALL SSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD ) 258 CALL SSYR( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD, 259 $ AB( KD+1, J+1 ), KLD ) 260 END IF 261 20 CONTINUE 262 ELSE 263* 264* Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m). 265* 266 DO 30 J = N, M + 1, -1 267* 268* Compute s(j,j) and test for non-positive-definiteness. 269* 270 AJJ = AB( 1, J ) 271 IF( AJJ.LE.ZERO ) 272 $ GO TO 50 273 AJJ = SQRT( AJJ ) 274 AB( 1, J ) = AJJ 275 KM = MIN( J-1, KD ) 276* 277* Compute elements j-km:j-1 of the j-th row and update the 278* trailing submatrix within the band. 279* 280 CALL SSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD ) 281 CALL SSYR( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD, 282 $ AB( 1, J-KM ), KLD ) 283 30 CONTINUE 284* 285* Factorize the updated submatrix A(1:m,1:m) as U**T*U. 286* 287 DO 40 J = 1, M 288* 289* Compute s(j,j) and test for non-positive-definiteness. 290* 291 AJJ = AB( 1, J ) 292 IF( AJJ.LE.ZERO ) 293 $ GO TO 50 294 AJJ = SQRT( AJJ ) 295 AB( 1, J ) = AJJ 296 KM = MIN( KD, M-J ) 297* 298* Compute elements j+1:j+km of the j-th column and update the 299* trailing submatrix within the band. 300* 301 IF( KM.GT.0 ) THEN 302 CALL SSCAL( KM, ONE / AJJ, AB( 2, J ), 1 ) 303 CALL SSYR( 'Lower', KM, -ONE, AB( 2, J ), 1, 304 $ AB( 1, J+1 ), KLD ) 305 END IF 306 40 CONTINUE 307 END IF 308 RETURN 309* 310 50 CONTINUE 311 INFO = J 312 RETURN 313* 314* End of SPBSTF 315* 316 END 317