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LAPACK 3.3.1
Linear Algebra PACKage
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LAPACK 3.3.1
Linear Algebra PACKage
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00001 SUBROUTINE ZPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, 00002 $ EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, 00003 $ WORK, RWORK, INFO ) 00004 * 00005 * -- LAPACK driver routine (version 3.3.1) -- 00006 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00007 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00008 * -- April 2011 -- 00009 * 00010 * .. Scalar Arguments .. 00011 CHARACTER EQUED, FACT, UPLO 00012 INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS 00013 DOUBLE PRECISION RCOND 00014 * .. 00015 * .. Array Arguments .. 00016 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * ) 00017 COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), 00018 $ WORK( * ), X( LDX, * ) 00019 * .. 00020 * 00021 * Purpose 00022 * ======= 00023 * 00024 * ZPBSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to 00025 * compute the solution to a complex system of linear equations 00026 * A * X = B, 00027 * where A is an N-by-N Hermitian positive definite band matrix and X 00028 * and B are N-by-NRHS matrices. 00029 * 00030 * Error bounds on the solution and a condition estimate are also 00031 * provided. 00032 * 00033 * Description 00034 * =========== 00035 * 00036 * The following steps are performed: 00037 * 00038 * 1. If FACT = 'E', real scaling factors are computed to equilibrate 00039 * the system: 00040 * diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B 00041 * Whether or not the system will be equilibrated depends on the 00042 * scaling of the matrix A, but if equilibration is used, A is 00043 * overwritten by diag(S)*A*diag(S) and B by diag(S)*B. 00044 * 00045 * 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to 00046 * factor the matrix A (after equilibration if FACT = 'E') as 00047 * A = U**H * U, if UPLO = 'U', or 00048 * A = L * L**H, if UPLO = 'L', 00049 * where U is an upper triangular band matrix, and L is a lower 00050 * triangular band matrix. 00051 * 00052 * 3. If the leading i-by-i principal minor is not positive definite, 00053 * then the routine returns with INFO = i. Otherwise, the factored 00054 * form of A is used to estimate the condition number of the matrix 00055 * A. If the reciprocal of the condition number is less than machine 00056 * precision, INFO = N+1 is returned as a warning, but the routine 00057 * still goes on to solve for X and compute error bounds as 00058 * described below. 00059 * 00060 * 4. The system of equations is solved for X using the factored form 00061 * of A. 00062 * 00063 * 5. Iterative refinement is applied to improve the computed solution 00064 * matrix and calculate error bounds and backward error estimates 00065 * for it. 00066 * 00067 * 6. If equilibration was used, the matrix X is premultiplied by 00068 * diag(S) so that it solves the original system before 00069 * equilibration. 00070 * 00071 * Arguments 00072 * ========= 00073 * 00074 * FACT (input) CHARACTER*1 00075 * Specifies whether or not the factored form of the matrix A is 00076 * supplied on entry, and if not, whether the matrix A should be 00077 * equilibrated before it is factored. 00078 * = 'F': On entry, AFB contains the factored form of A. 00079 * If EQUED = 'Y', the matrix A has been equilibrated 00080 * with scaling factors given by S. AB and AFB will not 00081 * be modified. 00082 * = 'N': The matrix A will be copied to AFB and factored. 00083 * = 'E': The matrix A will be equilibrated if necessary, then 00084 * copied to AFB and factored. 00085 * 00086 * UPLO (input) CHARACTER*1 00087 * = 'U': Upper triangle of A is stored; 00088 * = 'L': Lower triangle of A is stored. 00089 * 00090 * N (input) INTEGER 00091 * The number of linear equations, i.e., the order of the 00092 * matrix A. N >= 0. 00093 * 00094 * KD (input) INTEGER 00095 * The number of superdiagonals of the matrix A if UPLO = 'U', 00096 * or the number of subdiagonals if UPLO = 'L'. KD >= 0. 00097 * 00098 * NRHS (input) INTEGER 00099 * The number of right-hand sides, i.e., the number of columns 00100 * of the matrices B and X. NRHS >= 0. 00101 * 00102 * AB (input/output) COMPLEX*16 array, dimension (LDAB,N) 00103 * On entry, the upper or lower triangle of the Hermitian band 00104 * matrix A, stored in the first KD+1 rows of the array, except 00105 * if FACT = 'F' and EQUED = 'Y', then A must contain the 00106 * equilibrated matrix diag(S)*A*diag(S). The j-th column of A 00107 * is stored in the j-th column of the array AB as follows: 00108 * if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j; 00109 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD). 00110 * See below for further details. 00111 * 00112 * On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by 00113 * diag(S)*A*diag(S). 00114 * 00115 * LDAB (input) INTEGER 00116 * The leading dimension of the array A. LDAB >= KD+1. 00117 * 00118 * AFB (input or output) COMPLEX*16 array, dimension (LDAFB,N) 00119 * If FACT = 'F', then AFB is an input argument and on entry 00120 * contains the triangular factor U or L from the Cholesky 00121 * factorization A = U**H *U or A = L*L**H of the band matrix 00122 * A, in the same storage format as A (see AB). If EQUED = 'Y', 00123 * then AFB is the factored form of the equilibrated matrix A. 00124 * 00125 * If FACT = 'N', then AFB is an output argument and on exit 00126 * returns the triangular factor U or L from the Cholesky 00127 * factorization A = U**H *U or A = L*L**H. 00128 * 00129 * If FACT = 'E', then AFB is an output argument and on exit 00130 * returns the triangular factor U or L from the Cholesky 00131 * factorization A = U**H *U or A = L*L**H of the equilibrated 00132 * matrix A (see the description of A for the form of the 00133 * equilibrated matrix). 00134 * 00135 * LDAFB (input) INTEGER 00136 * The leading dimension of the array AFB. LDAFB >= KD+1. 00137 * 00138 * EQUED (input or output) CHARACTER*1 00139 * Specifies the form of equilibration that was done. 00140 * = 'N': No equilibration (always true if FACT = 'N'). 00141 * = 'Y': Equilibration was done, i.e., A has been replaced by 00142 * diag(S) * A * diag(S). 00143 * EQUED is an input argument if FACT = 'F'; otherwise, it is an 00144 * output argument. 00145 * 00146 * S (input or output) DOUBLE PRECISION array, dimension (N) 00147 * The scale factors for A; not accessed if EQUED = 'N'. S is 00148 * an input argument if FACT = 'F'; otherwise, S is an output 00149 * argument. If FACT = 'F' and EQUED = 'Y', each element of S 00150 * must be positive. 00151 * 00152 * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS) 00153 * On entry, the N-by-NRHS right hand side matrix B. 00154 * On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', 00155 * B is overwritten by diag(S) * B. 00156 * 00157 * LDB (input) INTEGER 00158 * The leading dimension of the array B. LDB >= max(1,N). 00159 * 00160 * X (output) COMPLEX*16 array, dimension (LDX,NRHS) 00161 * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to 00162 * the original system of equations. Note that if EQUED = 'Y', 00163 * A and B are modified on exit, and the solution to the 00164 * equilibrated system is inv(diag(S))*X. 00165 * 00166 * LDX (input) INTEGER 00167 * The leading dimension of the array X. LDX >= max(1,N). 00168 * 00169 * RCOND (output) DOUBLE PRECISION 00170 * The estimate of the reciprocal condition number of the matrix 00171 * A after equilibration (if done). If RCOND is less than the 00172 * machine precision (in particular, if RCOND = 0), the matrix 00173 * is singular to working precision. This condition is 00174 * indicated by a return code of INFO > 0. 00175 * 00176 * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 00177 * The estimated forward error bound for each solution vector 00178 * X(j) (the j-th column of the solution matrix X). 00179 * If XTRUE is the true solution corresponding to X(j), FERR(j) 00180 * is an estimated upper bound for the magnitude of the largest 00181 * element in (X(j) - XTRUE) divided by the magnitude of the 00182 * largest element in X(j). The estimate is as reliable as 00183 * the estimate for RCOND, and is almost always a slight 00184 * overestimate of the true error. 00185 * 00186 * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 00187 * The componentwise relative backward error of each solution 00188 * vector X(j) (i.e., the smallest relative change in 00189 * any element of A or B that makes X(j) an exact solution). 00190 * 00191 * WORK (workspace) COMPLEX*16 array, dimension (2*N) 00192 * 00193 * RWORK (workspace) DOUBLE PRECISION array, dimension (N) 00194 * 00195 * INFO (output) INTEGER 00196 * = 0: successful exit 00197 * < 0: if INFO = -i, the i-th argument had an illegal value 00198 * > 0: if INFO = i, and i is 00199 * <= N: the leading minor of order i of A is 00200 * not positive definite, so the factorization 00201 * could not be completed, and the solution has not 00202 * been computed. RCOND = 0 is returned. 00203 * = N+1: U is nonsingular, but RCOND is less than machine 00204 * precision, meaning that the matrix is singular 00205 * to working precision. Nevertheless, the 00206 * solution and error bounds are computed because 00207 * there are a number of situations where the 00208 * computed solution can be more accurate than the 00209 * value of RCOND would suggest. 00210 * 00211 * Further Details 00212 * =============== 00213 * 00214 * The band storage scheme is illustrated by the following example, when 00215 * N = 6, KD = 2, and UPLO = 'U': 00216 * 00217 * Two-dimensional storage of the Hermitian matrix A: 00218 * 00219 * a11 a12 a13 00220 * a22 a23 a24 00221 * a33 a34 a35 00222 * a44 a45 a46 00223 * a55 a56 00224 * (aij=conjg(aji)) a66 00225 * 00226 * Band storage of the upper triangle of A: 00227 * 00228 * * * a13 a24 a35 a46 00229 * * a12 a23 a34 a45 a56 00230 * a11 a22 a33 a44 a55 a66 00231 * 00232 * Similarly, if UPLO = 'L' the format of A is as follows: 00233 * 00234 * a11 a22 a33 a44 a55 a66 00235 * a21 a32 a43 a54 a65 * 00236 * a31 a42 a53 a64 * * 00237 * 00238 * Array elements marked * are not used by the routine. 00239 * 00240 * ===================================================================== 00241 * 00242 * .. Parameters .. 00243 DOUBLE PRECISION ZERO, ONE 00244 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00245 * .. 00246 * .. Local Scalars .. 00247 LOGICAL EQUIL, NOFACT, RCEQU, UPPER 00248 INTEGER I, INFEQU, J, J1, J2 00249 DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM 00250 * .. 00251 * .. External Functions .. 00252 LOGICAL LSAME 00253 DOUBLE PRECISION DLAMCH, ZLANHB 00254 EXTERNAL LSAME, DLAMCH, ZLANHB 00255 * .. 00256 * .. External Subroutines .. 00257 EXTERNAL XERBLA, ZCOPY, ZLACPY, ZLAQHB, ZPBCON, ZPBEQU, 00258 $ ZPBRFS, ZPBTRF, ZPBTRS 00259 * .. 00260 * .. Intrinsic Functions .. 00261 INTRINSIC MAX, MIN 00262 * .. 00263 * .. Executable Statements .. 00264 * 00265 INFO = 0 00266 NOFACT = LSAME( FACT, 'N' ) 00267 EQUIL = LSAME( FACT, 'E' ) 00268 UPPER = LSAME( UPLO, 'U' ) 00269 IF( NOFACT .OR. EQUIL ) THEN 00270 EQUED = 'N' 00271 RCEQU = .FALSE. 00272 ELSE 00273 RCEQU = LSAME( EQUED, 'Y' ) 00274 SMLNUM = DLAMCH( 'Safe minimum' ) 00275 BIGNUM = ONE / SMLNUM 00276 END IF 00277 * 00278 * Test the input parameters. 00279 * 00280 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) ) 00281 $ THEN 00282 INFO = -1 00283 ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00284 INFO = -2 00285 ELSE IF( N.LT.0 ) THEN 00286 INFO = -3 00287 ELSE IF( KD.LT.0 ) THEN 00288 INFO = -4 00289 ELSE IF( NRHS.LT.0 ) THEN 00290 INFO = -5 00291 ELSE IF( LDAB.LT.KD+1 ) THEN 00292 INFO = -7 00293 ELSE IF( LDAFB.LT.KD+1 ) THEN 00294 INFO = -9 00295 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 00296 $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 00297 INFO = -10 00298 ELSE 00299 IF( RCEQU ) THEN 00300 SMIN = BIGNUM 00301 SMAX = ZERO 00302 DO 10 J = 1, N 00303 SMIN = MIN( SMIN, S( J ) ) 00304 SMAX = MAX( SMAX, S( J ) ) 00305 10 CONTINUE 00306 IF( SMIN.LE.ZERO ) THEN 00307 INFO = -11 00308 ELSE IF( N.GT.0 ) THEN 00309 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) 00310 ELSE 00311 SCOND = ONE 00312 END IF 00313 END IF 00314 IF( INFO.EQ.0 ) THEN 00315 IF( LDB.LT.MAX( 1, N ) ) THEN 00316 INFO = -13 00317 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00318 INFO = -15 00319 END IF 00320 END IF 00321 END IF 00322 * 00323 IF( INFO.NE.0 ) THEN 00324 CALL XERBLA( 'ZPBSVX', -INFO ) 00325 RETURN 00326 END IF 00327 * 00328 IF( EQUIL ) THEN 00329 * 00330 * Compute row and column scalings to equilibrate the matrix A. 00331 * 00332 CALL ZPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFEQU ) 00333 IF( INFEQU.EQ.0 ) THEN 00334 * 00335 * Equilibrate the matrix. 00336 * 00337 CALL ZLAQHB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED ) 00338 RCEQU = LSAME( EQUED, 'Y' ) 00339 END IF 00340 END IF 00341 * 00342 * Scale the right-hand side. 00343 * 00344 IF( RCEQU ) THEN 00345 DO 30 J = 1, NRHS 00346 DO 20 I = 1, N 00347 B( I, J ) = S( I )*B( I, J ) 00348 20 CONTINUE 00349 30 CONTINUE 00350 END IF 00351 * 00352 IF( NOFACT .OR. EQUIL ) THEN 00353 * 00354 * Compute the Cholesky factorization A = U**H *U or A = L*L**H. 00355 * 00356 IF( UPPER ) THEN 00357 DO 40 J = 1, N 00358 J1 = MAX( J-KD, 1 ) 00359 CALL ZCOPY( J-J1+1, AB( KD+1-J+J1, J ), 1, 00360 $ AFB( KD+1-J+J1, J ), 1 ) 00361 40 CONTINUE 00362 ELSE 00363 DO 50 J = 1, N 00364 J2 = MIN( J+KD, N ) 00365 CALL ZCOPY( J2-J+1, AB( 1, J ), 1, AFB( 1, J ), 1 ) 00366 50 CONTINUE 00367 END IF 00368 * 00369 CALL ZPBTRF( UPLO, N, KD, AFB, LDAFB, INFO ) 00370 * 00371 * Return if INFO is non-zero. 00372 * 00373 IF( INFO.GT.0 )THEN 00374 RCOND = ZERO 00375 RETURN 00376 END IF 00377 END IF 00378 * 00379 * Compute the norm of the matrix A. 00380 * 00381 ANORM = ZLANHB( '1', UPLO, N, KD, AB, LDAB, RWORK ) 00382 * 00383 * Compute the reciprocal of the condition number of A. 00384 * 00385 CALL ZPBCON( UPLO, N, KD, AFB, LDAFB, ANORM, RCOND, WORK, RWORK, 00386 $ INFO ) 00387 * 00388 * Compute the solution matrix X. 00389 * 00390 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 00391 CALL ZPBTRS( UPLO, N, KD, NRHS, AFB, LDAFB, X, LDX, INFO ) 00392 * 00393 * Use iterative refinement to improve the computed solution and 00394 * compute error bounds and backward error estimates for it. 00395 * 00396 CALL ZPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X, 00397 $ LDX, FERR, BERR, WORK, RWORK, INFO ) 00398 * 00399 * Transform the solution matrix X to a solution of the original 00400 * system. 00401 * 00402 IF( RCEQU ) THEN 00403 DO 70 J = 1, NRHS 00404 DO 60 I = 1, N 00405 X( I, J ) = S( I )*X( I, J ) 00406 60 CONTINUE 00407 70 CONTINUE 00408 DO 80 J = 1, NRHS 00409 FERR( J ) = FERR( J ) / SCOND 00410 80 CONTINUE 00411 END IF 00412 * 00413 * Set INFO = N+1 if the matrix is singular to working precision. 00414 * 00415 IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) 00416 $ INFO = N + 1 00417 * 00418 RETURN 00419 * 00420 * End of ZPBSVX 00421 * 00422 END
1.7.3 
