1 SUBROUTINE ZPBSTF( 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*16 AB( LDAB, * ) 14* .. 15* 16* Purpose 17* ======= 18* 19* ZPBSTF computes a split Cholesky factorization of a complex 20* Hermitian positive definite band matrix A. 21* 22* This routine is designed to be used in conjunction with ZHBGST. 23* 24* The factorization has the form A = S**H*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) COMPLEX*16 array, dimension (LDAB,N) 48* On entry, the upper or lower triangle of the Hermitian 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**H*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; s12' denotes 99* conjg(s12); the diagonal elements of S are real. 100* 101* ===================================================================== 102* 103* .. Parameters .. 104 DOUBLE PRECISION ONE, ZERO 105 PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) 106* .. 107* .. Local Scalars .. 108 LOGICAL UPPER 109 INTEGER J, KLD, KM, M 110 DOUBLE PRECISION AJJ 111* .. 112* .. External Functions .. 113 LOGICAL LSAME 114 EXTERNAL LSAME 115* .. 116* .. External Subroutines .. 117 EXTERNAL XERBLA, ZDSCAL, ZHER, ZLACGV 118* .. 119* .. Intrinsic Functions .. 120 INTRINSIC DBLE, MAX, MIN, SQRT 121* .. 122* .. Executable Statements .. 123* 124* Test the input parameters. 125* 126 INFO = 0 127 UPPER = LSAME( UPLO, 'U' ) 128 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 129 INFO = -1 130 ELSE IF( N.LT.0 ) THEN 131 INFO = -2 132 ELSE IF( KD.LT.0 ) THEN 133 INFO = -3 134 ELSE IF( LDAB.LT.KD+1 ) THEN 135 INFO = -5 136 END IF 137 IF( INFO.NE.0 ) THEN 138 CALL XERBLA( 'ZPBSTF', -INFO ) 139 RETURN 140 END IF 141* 142* Quick return if possible 143* 144 IF( N.EQ.0 ) 145 $ RETURN 146* 147 KLD = MAX( 1, LDAB-1 ) 148* 149* Set the splitting point m. 150* 151 M = ( N+KD ) / 2 152* 153 IF( UPPER ) THEN 154* 155* Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m). 156* 157 DO 10 J = N, M + 1, -1 158* 159* Compute s(j,j) and test for non-positive-definiteness. 160* 161 AJJ = DBLE( AB( KD+1, J ) ) 162 IF( AJJ.LE.ZERO ) THEN 163 AB( KD+1, J ) = AJJ 164 GO TO 50 165 END IF 166 AJJ = SQRT( AJJ ) 167 AB( KD+1, J ) = AJJ 168 KM = MIN( J-1, KD ) 169* 170* Compute elements j-km:j-1 of the j-th column and update the 171* the leading submatrix within the band. 172* 173 CALL ZDSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 ) 174 CALL ZHER( 'Upper', KM, -ONE, AB( KD+1-KM, J ), 1, 175 $ AB( KD+1, J-KM ), KLD ) 176 10 CONTINUE 177* 178* Factorize the updated submatrix A(1:m,1:m) as U**H*U. 179* 180 DO 20 J = 1, M 181* 182* Compute s(j,j) and test for non-positive-definiteness. 183* 184 AJJ = DBLE( AB( KD+1, J ) ) 185 IF( AJJ.LE.ZERO ) THEN 186 AB( KD+1, J ) = AJJ 187 GO TO 50 188 END IF 189 AJJ = SQRT( AJJ ) 190 AB( KD+1, J ) = AJJ 191 KM = MIN( KD, M-J ) 192* 193* Compute elements j+1:j+km of the j-th row and update the 194* trailing submatrix within the band. 195* 196 IF( KM.GT.0 ) THEN 197 CALL ZDSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD ) 198 CALL ZLACGV( KM, AB( KD, J+1 ), KLD ) 199 CALL ZHER( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD, 200 $ AB( KD+1, J+1 ), KLD ) 201 CALL ZLACGV( KM, AB( KD, J+1 ), KLD ) 202 END IF 203 20 CONTINUE 204 ELSE 205* 206* Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m). 207* 208 DO 30 J = N, M + 1, -1 209* 210* Compute s(j,j) and test for non-positive-definiteness. 211* 212 AJJ = DBLE( AB( 1, J ) ) 213 IF( AJJ.LE.ZERO ) THEN 214 AB( 1, J ) = AJJ 215 GO TO 50 216 END IF 217 AJJ = SQRT( AJJ ) 218 AB( 1, J ) = AJJ 219 KM = MIN( J-1, KD ) 220* 221* Compute elements j-km:j-1 of the j-th row and update the 222* trailing submatrix within the band. 223* 224 CALL ZDSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD ) 225 CALL ZLACGV( KM, AB( KM+1, J-KM ), KLD ) 226 CALL ZHER( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD, 227 $ AB( 1, J-KM ), KLD ) 228 CALL ZLACGV( KM, AB( KM+1, J-KM ), KLD ) 229 30 CONTINUE 230* 231* Factorize the updated submatrix A(1:m,1:m) as U**H*U. 232* 233 DO 40 J = 1, M 234* 235* Compute s(j,j) and test for non-positive-definiteness. 236* 237 AJJ = DBLE( AB( 1, J ) ) 238 IF( AJJ.LE.ZERO ) THEN 239 AB( 1, J ) = AJJ 240 GO TO 50 241 END IF 242 AJJ = SQRT( AJJ ) 243 AB( 1, J ) = AJJ 244 KM = MIN( KD, M-J ) 245* 246* Compute elements j+1:j+km of the j-th column and update the 247* trailing submatrix within the band. 248* 249 IF( KM.GT.0 ) THEN 250 CALL ZDSCAL( KM, ONE / AJJ, AB( 2, J ), 1 ) 251 CALL ZHER( 'Lower', KM, -ONE, AB( 2, J ), 1, 252 $ AB( 1, J+1 ), KLD ) 253 END IF 254 40 CONTINUE 255 END IF 256 RETURN 257* 258 50 CONTINUE 259 INFO = J 260 RETURN 261* 262* End of ZPBSTF 263* 264 END 265