00001 SUBROUTINE ZPBSTF( 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 COMPLEX*16 AB( LDAB, * ) 00014 * .. 00015 * 00016 * Purpose 00017 * ======= 00018 * 00019 * ZPBSTF computes a split Cholesky factorization of a complex 00020 * Hermitian positive definite band matrix A. 00021 * 00022 * This routine is designed to be used in conjunction with ZHBGST. 00023 * 00024 * The factorization has the form A = S**H*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) COMPLEX*16 array, dimension (LDAB,N) 00048 * On entry, the upper or lower triangle of the Hermitian 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**H*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; s12' denotes 00099 * conjg(s12); the diagonal elements of S are real. 00100 * 00101 * ===================================================================== 00102 * 00103 * .. Parameters .. 00104 DOUBLE PRECISION ONE, ZERO 00105 PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) 00106 * .. 00107 * .. Local Scalars .. 00108 LOGICAL UPPER 00109 INTEGER J, KLD, KM, M 00110 DOUBLE PRECISION AJJ 00111 * .. 00112 * .. External Functions .. 00113 LOGICAL LSAME 00114 EXTERNAL LSAME 00115 * .. 00116 * .. External Subroutines .. 00117 EXTERNAL XERBLA, ZDSCAL, ZHER, ZLACGV 00118 * .. 00119 * .. Intrinsic Functions .. 00120 INTRINSIC DBLE, MAX, MIN, SQRT 00121 * .. 00122 * .. Executable Statements .. 00123 * 00124 * Test the input parameters. 00125 * 00126 INFO = 0 00127 UPPER = LSAME( UPLO, 'U' ) 00128 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00129 INFO = -1 00130 ELSE IF( N.LT.0 ) THEN 00131 INFO = -2 00132 ELSE IF( KD.LT.0 ) THEN 00133 INFO = -3 00134 ELSE IF( LDAB.LT.KD+1 ) THEN 00135 INFO = -5 00136 END IF 00137 IF( INFO.NE.0 ) THEN 00138 CALL XERBLA( 'ZPBSTF', -INFO ) 00139 RETURN 00140 END IF 00141 * 00142 * Quick return if possible 00143 * 00144 IF( N.EQ.0 ) 00145 $ RETURN 00146 * 00147 KLD = MAX( 1, LDAB-1 ) 00148 * 00149 * Set the splitting point m. 00150 * 00151 M = ( N+KD ) / 2 00152 * 00153 IF( UPPER ) THEN 00154 * 00155 * Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m). 00156 * 00157 DO 10 J = N, M + 1, -1 00158 * 00159 * Compute s(j,j) and test for non-positive-definiteness. 00160 * 00161 AJJ = DBLE( AB( KD+1, J ) ) 00162 IF( AJJ.LE.ZERO ) THEN 00163 AB( KD+1, J ) = AJJ 00164 GO TO 50 00165 END IF 00166 AJJ = SQRT( AJJ ) 00167 AB( KD+1, J ) = AJJ 00168 KM = MIN( J-1, KD ) 00169 * 00170 * Compute elements j-km:j-1 of the j-th column and update the 00171 * the leading submatrix within the band. 00172 * 00173 CALL ZDSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 ) 00174 CALL ZHER( 'Upper', KM, -ONE, AB( KD+1-KM, J ), 1, 00175 $ AB( KD+1, J-KM ), KLD ) 00176 10 CONTINUE 00177 * 00178 * Factorize the updated submatrix A(1:m,1:m) as U**H*U. 00179 * 00180 DO 20 J = 1, M 00181 * 00182 * Compute s(j,j) and test for non-positive-definiteness. 00183 * 00184 AJJ = DBLE( AB( KD+1, J ) ) 00185 IF( AJJ.LE.ZERO ) THEN 00186 AB( KD+1, J ) = AJJ 00187 GO TO 50 00188 END IF 00189 AJJ = SQRT( AJJ ) 00190 AB( KD+1, J ) = AJJ 00191 KM = MIN( KD, M-J ) 00192 * 00193 * Compute elements j+1:j+km of the j-th row and update the 00194 * trailing submatrix within the band. 00195 * 00196 IF( KM.GT.0 ) THEN 00197 CALL ZDSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD ) 00198 CALL ZLACGV( KM, AB( KD, J+1 ), KLD ) 00199 CALL ZHER( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD, 00200 $ AB( KD+1, J+1 ), KLD ) 00201 CALL ZLACGV( KM, AB( KD, J+1 ), KLD ) 00202 END IF 00203 20 CONTINUE 00204 ELSE 00205 * 00206 * Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m). 00207 * 00208 DO 30 J = N, M + 1, -1 00209 * 00210 * Compute s(j,j) and test for non-positive-definiteness. 00211 * 00212 AJJ = DBLE( AB( 1, J ) ) 00213 IF( AJJ.LE.ZERO ) THEN 00214 AB( 1, J ) = AJJ 00215 GO TO 50 00216 END IF 00217 AJJ = SQRT( AJJ ) 00218 AB( 1, J ) = AJJ 00219 KM = MIN( J-1, KD ) 00220 * 00221 * Compute elements j-km:j-1 of the j-th row and update the 00222 * trailing submatrix within the band. 00223 * 00224 CALL ZDSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD ) 00225 CALL ZLACGV( KM, AB( KM+1, J-KM ), KLD ) 00226 CALL ZHER( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD, 00227 $ AB( 1, J-KM ), KLD ) 00228 CALL ZLACGV( KM, AB( KM+1, J-KM ), KLD ) 00229 30 CONTINUE 00230 * 00231 * Factorize the updated submatrix A(1:m,1:m) as U**H*U. 00232 * 00233 DO 40 J = 1, M 00234 * 00235 * Compute s(j,j) and test for non-positive-definiteness. 00236 * 00237 AJJ = DBLE( AB( 1, J ) ) 00238 IF( AJJ.LE.ZERO ) THEN 00239 AB( 1, J ) = AJJ 00240 GO TO 50 00241 END IF 00242 AJJ = SQRT( AJJ ) 00243 AB( 1, J ) = AJJ 00244 KM = MIN( KD, M-J ) 00245 * 00246 * Compute elements j+1:j+km of the j-th column and update the 00247 * trailing submatrix within the band. 00248 * 00249 IF( KM.GT.0 ) THEN 00250 CALL ZDSCAL( KM, ONE / AJJ, AB( 2, J ), 1 ) 00251 CALL ZHER( 'Lower', KM, -ONE, AB( 2, J ), 1, 00252 $ AB( 1, J+1 ), KLD ) 00253 END IF 00254 40 CONTINUE 00255 END IF 00256 RETURN 00257 * 00258 50 CONTINUE 00259 INFO = J 00260 RETURN 00261 * 00262 * End of ZPBSTF 00263 * 00264 END
1.7.2