00001 SUBROUTINE DPBSTF( UPLO, N, KD, AB, LDAB, INFO ) 00002 * 00003 * -- LAPACK routine (version 3.2) -- 00004 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00005 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00006 * November 2006 00007 * 00008 * .. Scalar Arguments .. 00009 CHARACTER UPLO 00010 INTEGER INFO, KD, LDAB, N 00011 * .. 00012 * .. Array Arguments .. 00013 DOUBLE PRECISION AB( LDAB, * ) 00014 * .. 00015 * 00016 * Purpose 00017 * ======= 00018 * 00019 * DPBSTF computes a split Cholesky factorization of a real 00020 * symmetric positive definite band matrix A. 00021 * 00022 * This routine is designed to be used in conjunction with DSBGST. 00023 * 00024 * The factorization has the form A = S**T*S where S is a band matrix 00025 * of the same bandwidth as A and the following structure: 00026 * 00027 * S = ( U ) 00028 * ( M L ) 00029 * 00030 * where U is upper triangular of order m = (n+kd)/2, and L is lower 00031 * triangular of order n-m. 00032 * 00033 * Arguments 00034 * ========= 00035 * 00036 * UPLO (input) CHARACTER*1 00037 * = 'U': Upper triangle of A is stored; 00038 * = 'L': Lower triangle of A is stored. 00039 * 00040 * N (input) INTEGER 00041 * The order of the matrix A. N >= 0. 00042 * 00043 * KD (input) INTEGER 00044 * The number of superdiagonals of the matrix A if UPLO = 'U', 00045 * or the number of subdiagonals if UPLO = 'L'. KD >= 0. 00046 * 00047 * AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N) 00048 * On entry, the upper or lower triangle of the symmetric band 00049 * matrix A, stored in the first kd+1 rows of the array. The 00050 * j-th column of A is stored in the j-th column of the array AB 00051 * as follows: 00052 * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; 00053 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). 00054 * 00055 * On exit, if INFO = 0, the factor S from the split Cholesky 00056 * factorization A = S**T*S. See Further Details. 00057 * 00058 * LDAB (input) INTEGER 00059 * The leading dimension of the array AB. LDAB >= KD+1. 00060 * 00061 * INFO (output) INTEGER 00062 * = 0: successful exit 00063 * < 0: if INFO = -i, the i-th argument had an illegal value 00064 * > 0: if INFO = i, the factorization could not be completed, 00065 * because the updated element a(i,i) was negative; the 00066 * matrix A is not positive definite. 00067 * 00068 * Further Details 00069 * =============== 00070 * 00071 * The band storage scheme is illustrated by the following example, when 00072 * N = 7, KD = 2: 00073 * 00074 * S = ( s11 s12 s13 ) 00075 * ( s22 s23 s24 ) 00076 * ( s33 s34 ) 00077 * ( s44 ) 00078 * ( s53 s54 s55 ) 00079 * ( s64 s65 s66 ) 00080 * ( s75 s76 s77 ) 00081 * 00082 * If UPLO = 'U', the array AB holds: 00083 * 00084 * on entry: on exit: 00085 * 00086 * * * a13 a24 a35 a46 a57 * * s13 s24 s53 s64 s75 00087 * * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54 s65 s76 00088 * a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 00089 * 00090 * If UPLO = 'L', the array AB holds: 00091 * 00092 * on entry: on exit: 00093 * 00094 * a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 00095 * a21 a32 a43 a54 a65 a76 * s12 s23 s34 s54 s65 s76 * 00096 * a31 a42 a53 a64 a64 * * s13 s24 s53 s64 s75 * * 00097 * 00098 * Array elements marked * are not used by the routine. 00099 * 00100 * ===================================================================== 00101 * 00102 * .. Parameters .. 00103 DOUBLE PRECISION ONE, ZERO 00104 PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) 00105 * .. 00106 * .. Local Scalars .. 00107 LOGICAL UPPER 00108 INTEGER J, KLD, KM, M 00109 DOUBLE PRECISION AJJ 00110 * .. 00111 * .. External Functions .. 00112 LOGICAL LSAME 00113 EXTERNAL LSAME 00114 * .. 00115 * .. External Subroutines .. 00116 EXTERNAL DSCAL, DSYR, XERBLA 00117 * .. 00118 * .. Intrinsic Functions .. 00119 INTRINSIC MAX, MIN, SQRT 00120 * .. 00121 * .. Executable Statements .. 00122 * 00123 * Test the input parameters. 00124 * 00125 INFO = 0 00126 UPPER = LSAME( UPLO, 'U' ) 00127 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00128 INFO = -1 00129 ELSE IF( N.LT.0 ) THEN 00130 INFO = -2 00131 ELSE IF( KD.LT.0 ) THEN 00132 INFO = -3 00133 ELSE IF( LDAB.LT.KD+1 ) THEN 00134 INFO = -5 00135 END IF 00136 IF( INFO.NE.0 ) THEN 00137 CALL XERBLA( 'DPBSTF', -INFO ) 00138 RETURN 00139 END IF 00140 * 00141 * Quick return if possible 00142 * 00143 IF( N.EQ.0 ) 00144 $ RETURN 00145 * 00146 KLD = MAX( 1, LDAB-1 ) 00147 * 00148 * Set the splitting point m. 00149 * 00150 M = ( N+KD ) / 2 00151 * 00152 IF( UPPER ) THEN 00153 * 00154 * Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m). 00155 * 00156 DO 10 J = N, M + 1, -1 00157 * 00158 * Compute s(j,j) and test for non-positive-definiteness. 00159 * 00160 AJJ = AB( KD+1, J ) 00161 IF( AJJ.LE.ZERO ) 00162 $ GO TO 50 00163 AJJ = SQRT( AJJ ) 00164 AB( KD+1, J ) = AJJ 00165 KM = MIN( J-1, KD ) 00166 * 00167 * Compute elements j-km:j-1 of the j-th column and update the 00168 * the leading submatrix within the band. 00169 * 00170 CALL DSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 ) 00171 CALL DSYR( 'Upper', KM, -ONE, AB( KD+1-KM, J ), 1, 00172 $ AB( KD+1, J-KM ), KLD ) 00173 10 CONTINUE 00174 * 00175 * Factorize the updated submatrix A(1:m,1:m) as U**T*U. 00176 * 00177 DO 20 J = 1, M 00178 * 00179 * Compute s(j,j) and test for non-positive-definiteness. 00180 * 00181 AJJ = AB( KD+1, J ) 00182 IF( AJJ.LE.ZERO ) 00183 $ GO TO 50 00184 AJJ = SQRT( AJJ ) 00185 AB( KD+1, J ) = AJJ 00186 KM = MIN( KD, M-J ) 00187 * 00188 * Compute elements j+1:j+km of the j-th row and update the 00189 * trailing submatrix within the band. 00190 * 00191 IF( KM.GT.0 ) THEN 00192 CALL DSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD ) 00193 CALL DSYR( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD, 00194 $ AB( KD+1, J+1 ), KLD ) 00195 END IF 00196 20 CONTINUE 00197 ELSE 00198 * 00199 * Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m). 00200 * 00201 DO 30 J = N, M + 1, -1 00202 * 00203 * Compute s(j,j) and test for non-positive-definiteness. 00204 * 00205 AJJ = AB( 1, J ) 00206 IF( AJJ.LE.ZERO ) 00207 $ GO TO 50 00208 AJJ = SQRT( AJJ ) 00209 AB( 1, J ) = AJJ 00210 KM = MIN( J-1, KD ) 00211 * 00212 * Compute elements j-km:j-1 of the j-th row and update the 00213 * trailing submatrix within the band. 00214 * 00215 CALL DSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD ) 00216 CALL DSYR( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD, 00217 $ AB( 1, J-KM ), KLD ) 00218 30 CONTINUE 00219 * 00220 * Factorize the updated submatrix A(1:m,1:m) as U**T*U. 00221 * 00222 DO 40 J = 1, M 00223 * 00224 * Compute s(j,j) and test for non-positive-definiteness. 00225 * 00226 AJJ = AB( 1, J ) 00227 IF( AJJ.LE.ZERO ) 00228 $ GO TO 50 00229 AJJ = SQRT( AJJ ) 00230 AB( 1, J ) = AJJ 00231 KM = MIN( KD, M-J ) 00232 * 00233 * Compute elements j+1:j+km of the j-th column and update the 00234 * trailing submatrix within the band. 00235 * 00236 IF( KM.GT.0 ) THEN 00237 CALL DSCAL( KM, ONE / AJJ, AB( 2, J ), 1 ) 00238 CALL DSYR( 'Lower', KM, -ONE, AB( 2, J ), 1, 00239 $ AB( 1, J+1 ), KLD ) 00240 END IF 00241 40 CONTINUE 00242 END IF 00243 RETURN 00244 * 00245 50 CONTINUE 00246 INFO = J 00247 RETURN 00248 * 00249 * End of DPBSTF 00250 * 00251 END
1.7.2