00001 SUBROUTINE ZGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, 00002 $ DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, 00003 $ WORK, RWORK, INFO ) 00004 * 00005 * -- LAPACK routine (version 3.2) -- 00006 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00007 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00008 * November 2006 00009 * 00010 * .. Scalar Arguments .. 00011 CHARACTER FACT, TRANS 00012 INTEGER INFO, LDB, LDX, N, NRHS 00013 DOUBLE PRECISION RCOND 00014 * .. 00015 * .. Array Arguments .. 00016 INTEGER IPIV( * ) 00017 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ) 00018 COMPLEX*16 B( LDB, * ), D( * ), DF( * ), DL( * ), 00019 $ DLF( * ), DU( * ), DU2( * ), DUF( * ), 00020 $ WORK( * ), X( LDX, * ) 00021 * .. 00022 * 00023 * Purpose 00024 * ======= 00025 * 00026 * ZGTSVX uses the LU factorization to compute the solution to a complex 00027 * system of linear equations A * X = B, A**T * X = B, or A**H * X = B, 00028 * where A is a tridiagonal matrix of order N and X and B are N-by-NRHS 00029 * matrices. 00030 * 00031 * Error bounds on the solution and a condition estimate are also 00032 * provided. 00033 * 00034 * Description 00035 * =========== 00036 * 00037 * The following steps are performed: 00038 * 00039 * 1. If FACT = 'N', the LU decomposition is used to factor the matrix A 00040 * as A = L * U, where L is a product of permutation and unit lower 00041 * bidiagonal matrices and U is upper triangular with nonzeros in 00042 * only the main diagonal and first two superdiagonals. 00043 * 00044 * 2. If some U(i,i)=0, so that U is exactly singular, then the routine 00045 * returns with INFO = i. Otherwise, the factored form of A is used 00046 * to estimate the condition number of the matrix A. If the 00047 * reciprocal of the condition number is less than machine precision, 00048 * INFO = N+1 is returned as a warning, but the routine still goes on 00049 * to solve for X and compute error bounds as described below. 00050 * 00051 * 3. The system of equations is solved for X using the factored form 00052 * of A. 00053 * 00054 * 4. Iterative refinement is applied to improve the computed solution 00055 * matrix and calculate error bounds and backward error estimates 00056 * for it. 00057 * 00058 * Arguments 00059 * ========= 00060 * 00061 * FACT (input) CHARACTER*1 00062 * Specifies whether or not the factored form of A has been 00063 * supplied on entry. 00064 * = 'F': DLF, DF, DUF, DU2, and IPIV contain the factored form 00065 * of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV will not 00066 * be modified. 00067 * = 'N': The matrix will be copied to DLF, DF, and DUF 00068 * and factored. 00069 * 00070 * TRANS (input) CHARACTER*1 00071 * Specifies the form of the system of equations: 00072 * = 'N': A * X = B (No transpose) 00073 * = 'T': A**T * X = B (Transpose) 00074 * = 'C': A**H * X = B (Conjugate transpose) 00075 * 00076 * N (input) INTEGER 00077 * The order of the matrix A. N >= 0. 00078 * 00079 * NRHS (input) INTEGER 00080 * The number of right hand sides, i.e., the number of columns 00081 * of the matrix B. NRHS >= 0. 00082 * 00083 * DL (input) COMPLEX*16 array, dimension (N-1) 00084 * The (n-1) subdiagonal elements of A. 00085 * 00086 * D (input) COMPLEX*16 array, dimension (N) 00087 * The n diagonal elements of A. 00088 * 00089 * DU (input) COMPLEX*16 array, dimension (N-1) 00090 * The (n-1) superdiagonal elements of A. 00091 * 00092 * DLF (input or output) COMPLEX*16 array, dimension (N-1) 00093 * If FACT = 'F', then DLF is an input argument and on entry 00094 * contains the (n-1) multipliers that define the matrix L from 00095 * the LU factorization of A as computed by ZGTTRF. 00096 * 00097 * If FACT = 'N', then DLF is an output argument and on exit 00098 * contains the (n-1) multipliers that define the matrix L from 00099 * the LU factorization of A. 00100 * 00101 * DF (input or output) COMPLEX*16 array, dimension (N) 00102 * If FACT = 'F', then DF is an input argument and on entry 00103 * contains the n diagonal elements of the upper triangular 00104 * matrix U from the LU factorization of A. 00105 * 00106 * If FACT = 'N', then DF is an output argument and on exit 00107 * contains the n diagonal elements of the upper triangular 00108 * matrix U from the LU factorization of A. 00109 * 00110 * DUF (input or output) COMPLEX*16 array, dimension (N-1) 00111 * If FACT = 'F', then DUF is an input argument and on entry 00112 * contains the (n-1) elements of the first superdiagonal of U. 00113 * 00114 * If FACT = 'N', then DUF is an output argument and on exit 00115 * contains the (n-1) elements of the first superdiagonal of U. 00116 * 00117 * DU2 (input or output) COMPLEX*16 array, dimension (N-2) 00118 * If FACT = 'F', then DU2 is an input argument and on entry 00119 * contains the (n-2) elements of the second superdiagonal of 00120 * U. 00121 * 00122 * If FACT = 'N', then DU2 is an output argument and on exit 00123 * contains the (n-2) elements of the second superdiagonal of 00124 * U. 00125 * 00126 * IPIV (input or output) INTEGER array, dimension (N) 00127 * If FACT = 'F', then IPIV is an input argument and on entry 00128 * contains the pivot indices from the LU factorization of A as 00129 * computed by ZGTTRF. 00130 * 00131 * If FACT = 'N', then IPIV is an output argument and on exit 00132 * contains the pivot indices from the LU factorization of A; 00133 * row i of the matrix was interchanged with row IPIV(i). 00134 * IPIV(i) will always be either i or i+1; IPIV(i) = i indicates 00135 * a row interchange was not required. 00136 * 00137 * B (input) COMPLEX*16 array, dimension (LDB,NRHS) 00138 * The N-by-NRHS right hand side matrix B. 00139 * 00140 * LDB (input) INTEGER 00141 * The leading dimension of the array B. LDB >= max(1,N). 00142 * 00143 * X (output) COMPLEX*16 array, dimension (LDX,NRHS) 00144 * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X. 00145 * 00146 * LDX (input) INTEGER 00147 * The leading dimension of the array X. LDX >= max(1,N). 00148 * 00149 * RCOND (output) DOUBLE PRECISION 00150 * The estimate of the reciprocal condition number of the matrix 00151 * A. If RCOND is less than the machine precision (in 00152 * particular, if RCOND = 0), the matrix is singular to working 00153 * precision. This condition is indicated by a return code of 00154 * INFO > 0. 00155 * 00156 * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 00157 * The estimated forward error bound for each solution vector 00158 * X(j) (the j-th column of the solution matrix X). 00159 * If XTRUE is the true solution corresponding to X(j), FERR(j) 00160 * is an estimated upper bound for the magnitude of the largest 00161 * element in (X(j) - XTRUE) divided by the magnitude of the 00162 * largest element in X(j). The estimate is as reliable as 00163 * the estimate for RCOND, and is almost always a slight 00164 * overestimate of the true error. 00165 * 00166 * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 00167 * The componentwise relative backward error of each solution 00168 * vector X(j) (i.e., the smallest relative change in 00169 * any element of A or B that makes X(j) an exact solution). 00170 * 00171 * WORK (workspace) COMPLEX*16 array, dimension (2*N) 00172 * 00173 * RWORK (workspace) DOUBLE PRECISION array, dimension (N) 00174 * 00175 * INFO (output) INTEGER 00176 * = 0: successful exit 00177 * < 0: if INFO = -i, the i-th argument had an illegal value 00178 * > 0: if INFO = i, and i is 00179 * <= N: U(i,i) is exactly zero. The factorization 00180 * has not been completed unless i = N, but the 00181 * factor U is exactly singular, so the solution 00182 * and error bounds could not be computed. 00183 * RCOND = 0 is returned. 00184 * = N+1: U is nonsingular, but RCOND is less than machine 00185 * precision, meaning that the matrix is singular 00186 * to working precision. Nevertheless, the 00187 * solution and error bounds are computed because 00188 * there are a number of situations where the 00189 * computed solution can be more accurate than the 00190 * value of RCOND would suggest. 00191 * 00192 * ===================================================================== 00193 * 00194 * .. Parameters .. 00195 DOUBLE PRECISION ZERO 00196 PARAMETER ( ZERO = 0.0D+0 ) 00197 * .. 00198 * .. Local Scalars .. 00199 LOGICAL NOFACT, NOTRAN 00200 CHARACTER NORM 00201 DOUBLE PRECISION ANORM 00202 * .. 00203 * .. External Functions .. 00204 LOGICAL LSAME 00205 DOUBLE PRECISION DLAMCH, ZLANGT 00206 EXTERNAL LSAME, DLAMCH, ZLANGT 00207 * .. 00208 * .. External Subroutines .. 00209 EXTERNAL XERBLA, ZCOPY, ZGTCON, ZGTRFS, ZGTTRF, ZGTTRS, 00210 $ ZLACPY 00211 * .. 00212 * .. Intrinsic Functions .. 00213 INTRINSIC MAX 00214 * .. 00215 * .. Executable Statements .. 00216 * 00217 INFO = 0 00218 NOFACT = LSAME( FACT, 'N' ) 00219 NOTRAN = LSAME( TRANS, 'N' ) 00220 IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN 00221 INFO = -1 00222 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. 00223 $ LSAME( TRANS, 'C' ) ) THEN 00224 INFO = -2 00225 ELSE IF( N.LT.0 ) THEN 00226 INFO = -3 00227 ELSE IF( NRHS.LT.0 ) THEN 00228 INFO = -4 00229 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00230 INFO = -14 00231 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00232 INFO = -16 00233 END IF 00234 IF( INFO.NE.0 ) THEN 00235 CALL XERBLA( 'ZGTSVX', -INFO ) 00236 RETURN 00237 END IF 00238 * 00239 IF( NOFACT ) THEN 00240 * 00241 * Compute the LU factorization of A. 00242 * 00243 CALL ZCOPY( N, D, 1, DF, 1 ) 00244 IF( N.GT.1 ) THEN 00245 CALL ZCOPY( N-1, DL, 1, DLF, 1 ) 00246 CALL ZCOPY( N-1, DU, 1, DUF, 1 ) 00247 END IF 00248 CALL ZGTTRF( N, DLF, DF, DUF, DU2, IPIV, INFO ) 00249 * 00250 * Return if INFO is non-zero. 00251 * 00252 IF( INFO.GT.0 )THEN 00253 RCOND = ZERO 00254 RETURN 00255 END IF 00256 END IF 00257 * 00258 * Compute the norm of the matrix A. 00259 * 00260 IF( NOTRAN ) THEN 00261 NORM = '1' 00262 ELSE 00263 NORM = 'I' 00264 END IF 00265 ANORM = ZLANGT( NORM, N, DL, D, DU ) 00266 * 00267 * Compute the reciprocal of the condition number of A. 00268 * 00269 CALL ZGTCON( NORM, N, DLF, DF, DUF, DU2, IPIV, ANORM, RCOND, WORK, 00270 $ INFO ) 00271 * 00272 * Compute the solution vectors X. 00273 * 00274 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 00275 CALL ZGTTRS( TRANS, N, NRHS, DLF, DF, DUF, DU2, IPIV, X, LDX, 00276 $ INFO ) 00277 * 00278 * Use iterative refinement to improve the computed solutions and 00279 * compute error bounds and backward error estimates for them. 00280 * 00281 CALL ZGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, 00282 $ B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO ) 00283 * 00284 * Set INFO = N+1 if the matrix is singular to working precision. 00285 * 00286 IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) 00287 $ INFO = N + 1 00288 * 00289 RETURN 00290 * 00291 * End of ZGTSVX 00292 * 00293 END
1.7.2