1 SUBROUTINE SPBSTF( 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 REAL AB( LDAB, * ) 14* .. 15* 16* Purpose 17* ======= 18* 19* SPBSTF computes a split Cholesky factorization of a real 20* symmetric positive definite band matrix A. 21* 22* This routine is designed to be used in conjunction with SSBGST. 23* 24* The factorization has the form A = S**T*S where S is a band matrix 25* of the same bandwidth as A and the following structure: 26* 27* S = ( U ) 28* ( M L ) 29* 30* where U is upper triangular of order m = (n+kd)/2, and L is lower 31* triangular of order n-m. 32* 33* Arguments 34* ========= 35* 36* UPLO (input) CHARACTER*1 37* = 'U': Upper triangle of A is stored; 38* = 'L': Lower triangle of A is stored. 39* 40* N (input) INTEGER 41* The order of the matrix A. N >= 0. 42* 43* KD (input) INTEGER 44* The number of superdiagonals of the matrix A if UPLO = 'U', 45* or the number of subdiagonals if UPLO = 'L'. KD >= 0. 46* 47* AB (input/output) REAL array, dimension (LDAB,N) 48* On entry, the upper or lower triangle of the symmetric band 49* matrix A, stored in the first kd+1 rows of the array. The 50* j-th column of A is stored in the j-th column of the array AB 51* as follows: 52* if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; 53* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). 54* 55* On exit, if INFO = 0, the factor S from the split Cholesky 56* factorization A = S**T*S. See Further Details. 57* 58* LDAB (input) INTEGER 59* The leading dimension of the array AB. LDAB >= KD+1. 60* 61* INFO (output) INTEGER 62* = 0: successful exit 63* < 0: if INFO = -i, the i-th argument had an illegal value 64* > 0: if INFO = i, the factorization could not be completed, 65* because the updated element a(i,i) was negative; the 66* matrix A is not positive definite. 67* 68* Further Details 69* =============== 70* 71* The band storage scheme is illustrated by the following example, when 72* N = 7, KD = 2: 73* 74* S = ( s11 s12 s13 ) 75* ( s22 s23 s24 ) 76* ( s33 s34 ) 77* ( s44 ) 78* ( s53 s54 s55 ) 79* ( s64 s65 s66 ) 80* ( s75 s76 s77 ) 81* 82* If UPLO = 'U', the array AB holds: 83* 84* on entry: on exit: 85* 86* * * a13 a24 a35 a46 a57 * * s13 s24 s53 s64 s75 87* * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54 s65 s76 88* a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 89* 90* If UPLO = 'L', the array AB holds: 91* 92* on entry: on exit: 93* 94* a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 95* a21 a32 a43 a54 a65 a76 * s12 s23 s34 s54 s65 s76 * 96* a31 a42 a53 a64 a64 * * s13 s24 s53 s64 s75 * * 97* 98* Array elements marked * are not used by the routine. 99* 100* ===================================================================== 101* 102* .. Parameters .. 103 REAL ONE, ZERO 104 PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 ) 105* .. 106* .. Local Scalars .. 107 LOGICAL UPPER 108 INTEGER J, KLD, KM, M 109 REAL AJJ 110* .. 111* .. External Functions .. 112 LOGICAL LSAME 113 EXTERNAL LSAME 114* .. 115* .. External Subroutines .. 116 EXTERNAL SSCAL, SSYR, XERBLA 117* .. 118* .. Intrinsic Functions .. 119 INTRINSIC MAX, MIN, SQRT 120* .. 121* .. Executable Statements .. 122* 123* Test the input parameters. 124* 125 INFO = 0 126 UPPER = LSAME( UPLO, 'U' ) 127 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 128 INFO = -1 129 ELSE IF( N.LT.0 ) THEN 130 INFO = -2 131 ELSE IF( KD.LT.0 ) THEN 132 INFO = -3 133 ELSE IF( LDAB.LT.KD+1 ) THEN 134 INFO = -5 135 END IF 136 IF( INFO.NE.0 ) THEN 137 CALL XERBLA( 'SPBSTF', -INFO ) 138 RETURN 139 END IF 140* 141* Quick return if possible 142* 143 IF( N.EQ.0 ) 144 $ RETURN 145* 146 KLD = MAX( 1, LDAB-1 ) 147* 148* Set the splitting point m. 149* 150 M = ( N+KD ) / 2 151* 152 IF( UPPER ) THEN 153* 154* Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m). 155* 156 DO 10 J = N, M + 1, -1 157* 158* Compute s(j,j) and test for non-positive-definiteness. 159* 160 AJJ = AB( KD+1, J ) 161 IF( AJJ.LE.ZERO ) 162 $ GO TO 50 163 AJJ = SQRT( AJJ ) 164 AB( KD+1, J ) = AJJ 165 KM = MIN( J-1, KD ) 166* 167* Compute elements j-km:j-1 of the j-th column and update the 168* the leading submatrix within the band. 169* 170 CALL SSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 ) 171 CALL SSYR( 'Upper', KM, -ONE, AB( KD+1-KM, J ), 1, 172 $ AB( KD+1, J-KM ), KLD ) 173 10 CONTINUE 174* 175* Factorize the updated submatrix A(1:m,1:m) as U**T*U. 176* 177 DO 20 J = 1, M 178* 179* Compute s(j,j) and test for non-positive-definiteness. 180* 181 AJJ = AB( KD+1, J ) 182 IF( AJJ.LE.ZERO ) 183 $ GO TO 50 184 AJJ = SQRT( AJJ ) 185 AB( KD+1, J ) = AJJ 186 KM = MIN( KD, M-J ) 187* 188* Compute elements j+1:j+km of the j-th row and update the 189* trailing submatrix within the band. 190* 191 IF( KM.GT.0 ) THEN 192 CALL SSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD ) 193 CALL SSYR( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD, 194 $ AB( KD+1, J+1 ), KLD ) 195 END IF 196 20 CONTINUE 197 ELSE 198* 199* Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m). 200* 201 DO 30 J = N, M + 1, -1 202* 203* Compute s(j,j) and test for non-positive-definiteness. 204* 205 AJJ = AB( 1, J ) 206 IF( AJJ.LE.ZERO ) 207 $ GO TO 50 208 AJJ = SQRT( AJJ ) 209 AB( 1, J ) = AJJ 210 KM = MIN( J-1, KD ) 211* 212* Compute elements j-km:j-1 of the j-th row and update the 213* trailing submatrix within the band. 214* 215 CALL SSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD ) 216 CALL SSYR( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD, 217 $ AB( 1, J-KM ), KLD ) 218 30 CONTINUE 219* 220* Factorize the updated submatrix A(1:m,1:m) as U**T*U. 221* 222 DO 40 J = 1, M 223* 224* Compute s(j,j) and test for non-positive-definiteness. 225* 226 AJJ = AB( 1, J ) 227 IF( AJJ.LE.ZERO ) 228 $ GO TO 50 229 AJJ = SQRT( AJJ ) 230 AB( 1, J ) = AJJ 231 KM = MIN( KD, M-J ) 232* 233* Compute elements j+1:j+km of the j-th column and update the 234* trailing submatrix within the band. 235* 236 IF( KM.GT.0 ) THEN 237 CALL SSCAL( KM, ONE / AJJ, AB( 2, J ), 1 ) 238 CALL SSYR( 'Lower', KM, -ONE, AB( 2, J ), 1, 239 $ AB( 1, J+1 ), KLD ) 240 END IF 241 40 CONTINUE 242 END IF 243 RETURN 244* 245 50 CONTINUE 246 INFO = J 247 RETURN 248* 249* End of SPBSTF 250* 251 END 252