00001 /* cppsvx.f -- translated by f2c (version 20061008). 00002 You must link the resulting object file with libf2c: 00003 on Microsoft Windows system, link with libf2c.lib; 00004 on Linux or Unix systems, link with .../path/to/libf2c.a -lm 00005 or, if you install libf2c.a in a standard place, with -lf2c -lm 00006 -- in that order, at the end of the command line, as in 00007 cc *.o -lf2c -lm 00008 Source for libf2c is in /netlib/f2c/libf2c.zip, e.g., 00009 00010 http://www.netlib.org/f2c/libf2c.zip 00011 */ 00012 00013 #include "f2c.h" 00014 #include "blaswrap.h" 00015 00016 /* Table of constant values */ 00017 00018 static integer c__1 = 1; 00019 00020 /* Subroutine */ int cppsvx_(char *fact, char *uplo, integer *n, integer * 00021 nrhs, complex *ap, complex *afp, char *equed, real *s, complex *b, 00022 integer *ldb, complex *x, integer *ldx, real *rcond, real *ferr, real 00023 *berr, complex *work, real *rwork, integer *info) 00024 { 00025 /* System generated locals */ 00026 integer b_dim1, b_offset, x_dim1, x_offset, i__1, i__2, i__3, i__4, i__5; 00027 real r__1, r__2; 00028 complex q__1; 00029 00030 /* Local variables */ 00031 integer i__, j; 00032 real amax, smin, smax; 00033 extern logical lsame_(char *, char *); 00034 real scond, anorm; 00035 extern /* Subroutine */ int ccopy_(integer *, complex *, integer *, 00036 complex *, integer *); 00037 logical equil, rcequ; 00038 extern doublereal clanhp_(char *, char *, integer *, complex *, real *), slamch_(char *); 00039 extern /* Subroutine */ int claqhp_(char *, integer *, complex *, real *, 00040 real *, real *, char *); 00041 logical nofact; 00042 extern /* Subroutine */ int clacpy_(char *, integer *, integer *, complex 00043 *, integer *, complex *, integer *), xerbla_(char *, 00044 integer *); 00045 real bignum; 00046 extern /* Subroutine */ int cppcon_(char *, integer *, complex *, real *, 00047 real *, complex *, real *, integer *); 00048 integer infequ; 00049 extern /* Subroutine */ int cppequ_(char *, integer *, complex *, real *, 00050 real *, real *, integer *), cpprfs_(char *, integer *, 00051 integer *, complex *, complex *, complex *, integer *, complex *, 00052 integer *, real *, real *, complex *, real *, integer *), 00053 cpptrf_(char *, integer *, complex *, integer *); 00054 real smlnum; 00055 extern /* Subroutine */ int cpptrs_(char *, integer *, integer *, complex 00056 *, complex *, integer *, integer *); 00057 00058 00059 /* -- LAPACK driver routine (version 3.2) -- */ 00060 /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ 00061 /* November 2006 */ 00062 00063 /* .. Scalar Arguments .. */ 00064 /* .. */ 00065 /* .. Array Arguments .. */ 00066 /* .. */ 00067 00068 /* Purpose */ 00069 /* ======= */ 00070 00071 /* CPPSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to */ 00072 /* compute the solution to a complex system of linear equations */ 00073 /* A * X = B, */ 00074 /* where A is an N-by-N Hermitian positive definite matrix stored in */ 00075 /* packed format and X and B are N-by-NRHS matrices. */ 00076 00077 /* Error bounds on the solution and a condition estimate are also */ 00078 /* provided. */ 00079 00080 /* Description */ 00081 /* =========== */ 00082 00083 /* The following steps are performed: */ 00084 00085 /* 1. If FACT = 'E', real scaling factors are computed to equilibrate */ 00086 /* the system: */ 00087 /* diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B */ 00088 /* Whether or not the system will be equilibrated depends on the */ 00089 /* scaling of the matrix A, but if equilibration is used, A is */ 00090 /* overwritten by diag(S)*A*diag(S) and B by diag(S)*B. */ 00091 00092 /* 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to */ 00093 /* factor the matrix A (after equilibration if FACT = 'E') as */ 00094 /* A = U'* U , if UPLO = 'U', or */ 00095 /* A = L * L', if UPLO = 'L', */ 00096 /* where U is an upper triangular matrix, L is a lower triangular */ 00097 /* matrix, and ' indicates conjugate transpose. */ 00098 00099 /* 3. If the leading i-by-i principal minor is not positive definite, */ 00100 /* then the routine returns with INFO = i. Otherwise, the factored */ 00101 /* form of A is used to estimate the condition number of the matrix */ 00102 /* A. If the reciprocal of the condition number is less than machine */ 00103 /* precision, INFO = N+1 is returned as a warning, but the routine */ 00104 /* still goes on to solve for X and compute error bounds as */ 00105 /* described below. */ 00106 00107 /* 4. The system of equations is solved for X using the factored form */ 00108 /* of A. */ 00109 00110 /* 5. Iterative refinement is applied to improve the computed solution */ 00111 /* matrix and calculate error bounds and backward error estimates */ 00112 /* for it. */ 00113 00114 /* 6. If equilibration was used, the matrix X is premultiplied by */ 00115 /* diag(S) so that it solves the original system before */ 00116 /* equilibration. */ 00117 00118 /* Arguments */ 00119 /* ========= */ 00120 00121 /* FACT (input) CHARACTER*1 */ 00122 /* Specifies whether or not the factored form of the matrix A is */ 00123 /* supplied on entry, and if not, whether the matrix A should be */ 00124 /* equilibrated before it is factored. */ 00125 /* = 'F': On entry, AFP contains the factored form of A. */ 00126 /* If EQUED = 'Y', the matrix A has been equilibrated */ 00127 /* with scaling factors given by S. AP and AFP will not */ 00128 /* be modified. */ 00129 /* = 'N': The matrix A will be copied to AFP and factored. */ 00130 /* = 'E': The matrix A will be equilibrated if necessary, then */ 00131 /* copied to AFP and factored. */ 00132 00133 /* UPLO (input) CHARACTER*1 */ 00134 /* = 'U': Upper triangle of A is stored; */ 00135 /* = 'L': Lower triangle of A is stored. */ 00136 00137 /* N (input) INTEGER */ 00138 /* The number of linear equations, i.e., the order of the */ 00139 /* matrix A. N >= 0. */ 00140 00141 /* NRHS (input) INTEGER */ 00142 /* The number of right hand sides, i.e., the number of columns */ 00143 /* of the matrices B and X. NRHS >= 0. */ 00144 00145 /* AP (input/output) COMPLEX array, dimension (N*(N+1)/2) */ 00146 /* On entry, the upper or lower triangle of the Hermitian matrix */ 00147 /* A, packed columnwise in a linear array, except if FACT = 'F' */ 00148 /* and EQUED = 'Y', then A must contain the equilibrated matrix */ 00149 /* diag(S)*A*diag(S). The j-th column of A is stored in the */ 00150 /* array AP as follows: */ 00151 /* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; */ 00152 /* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. */ 00153 /* See below for further details. A is not modified if */ 00154 /* FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. */ 00155 00156 /* On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by */ 00157 /* diag(S)*A*diag(S). */ 00158 00159 /* AFP (input or output) COMPLEX array, dimension (N*(N+1)/2) */ 00160 /* If FACT = 'F', then AFP is an input argument and on entry */ 00161 /* contains the triangular factor U or L from the Cholesky */ 00162 /* factorization A = U**H*U or A = L*L**H, in the same storage */ 00163 /* format as A. If EQUED .ne. 'N', then AFP is the factored */ 00164 /* form of the equilibrated matrix A. */ 00165 00166 /* If FACT = 'N', then AFP is an output argument and on exit */ 00167 /* returns the triangular factor U or L from the Cholesky */ 00168 /* factorization A = U**H*U or A = L*L**H of the original */ 00169 /* matrix A. */ 00170 00171 /* If FACT = 'E', then AFP is an output argument and on exit */ 00172 /* returns the triangular factor U or L from the Cholesky */ 00173 /* factorization A = U**H*U or A = L*L**H of the equilibrated */ 00174 /* matrix A (see the description of AP for the form of the */ 00175 /* equilibrated matrix). */ 00176 00177 /* EQUED (input or output) CHARACTER*1 */ 00178 /* Specifies the form of equilibration that was done. */ 00179 /* = 'N': No equilibration (always true if FACT = 'N'). */ 00180 /* = 'Y': Equilibration was done, i.e., A has been replaced by */ 00181 /* diag(S) * A * diag(S). */ 00182 /* EQUED is an input argument if FACT = 'F'; otherwise, it is an */ 00183 /* output argument. */ 00184 00185 /* S (input or output) REAL array, dimension (N) */ 00186 /* The scale factors for A; not accessed if EQUED = 'N'. S is */ 00187 /* an input argument if FACT = 'F'; otherwise, S is an output */ 00188 /* argument. If FACT = 'F' and EQUED = 'Y', each element of S */ 00189 /* must be positive. */ 00190 00191 /* B (input/output) COMPLEX array, dimension (LDB,NRHS) */ 00192 /* On entry, the N-by-NRHS right hand side matrix B. */ 00193 /* On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', */ 00194 /* B is overwritten by diag(S) * B. */ 00195 00196 /* LDB (input) INTEGER */ 00197 /* The leading dimension of the array B. LDB >= max(1,N). */ 00198 00199 /* X (output) COMPLEX array, dimension (LDX,NRHS) */ 00200 /* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to */ 00201 /* the original system of equations. Note that if EQUED = 'Y', */ 00202 /* A and B are modified on exit, and the solution to the */ 00203 /* equilibrated system is inv(diag(S))*X. */ 00204 00205 /* LDX (input) INTEGER */ 00206 /* The leading dimension of the array X. LDX >= max(1,N). */ 00207 00208 /* RCOND (output) REAL */ 00209 /* The estimate of the reciprocal condition number of the matrix */ 00210 /* A after equilibration (if done). If RCOND is less than the */ 00211 /* machine precision (in particular, if RCOND = 0), the matrix */ 00212 /* is singular to working precision. This condition is */ 00213 /* indicated by a return code of INFO > 0. */ 00214 00215 /* FERR (output) REAL array, dimension (NRHS) */ 00216 /* The estimated forward error bound for each solution vector */ 00217 /* X(j) (the j-th column of the solution matrix X). */ 00218 /* If XTRUE is the true solution corresponding to X(j), FERR(j) */ 00219 /* is an estimated upper bound for the magnitude of the largest */ 00220 /* element in (X(j) - XTRUE) divided by the magnitude of the */ 00221 /* largest element in X(j). The estimate is as reliable as */ 00222 /* the estimate for RCOND, and is almost always a slight */ 00223 /* overestimate of the true error. */ 00224 00225 /* BERR (output) REAL array, dimension (NRHS) */ 00226 /* The componentwise relative backward error of each solution */ 00227 /* vector X(j) (i.e., the smallest relative change in */ 00228 /* any element of A or B that makes X(j) an exact solution). */ 00229 00230 /* WORK (workspace) COMPLEX array, dimension (2*N) */ 00231 00232 /* RWORK (workspace) REAL array, dimension (N) */ 00233 00234 /* INFO (output) INTEGER */ 00235 /* = 0: successful exit */ 00236 /* < 0: if INFO = -i, the i-th argument had an illegal value */ 00237 /* > 0: if INFO = i, and i is */ 00238 /* <= N: the leading minor of order i of A is */ 00239 /* not positive definite, so the factorization */ 00240 /* could not be completed, and the solution has not */ 00241 /* been computed. RCOND = 0 is returned. */ 00242 /* = N+1: U is nonsingular, but RCOND is less than machine */ 00243 /* precision, meaning that the matrix is singular */ 00244 /* to working precision. Nevertheless, the */ 00245 /* solution and error bounds are computed because */ 00246 /* there are a number of situations where the */ 00247 /* computed solution can be more accurate than the */ 00248 /* value of RCOND would suggest. */ 00249 00250 /* Further Details */ 00251 /* =============== */ 00252 00253 /* The packed storage scheme is illustrated by the following example */ 00254 /* when N = 4, UPLO = 'U': */ 00255 00256 /* Two-dimensional storage of the Hermitian matrix A: */ 00257 00258 /* a11 a12 a13 a14 */ 00259 /* a22 a23 a24 */ 00260 /* a33 a34 (aij = conjg(aji)) */ 00261 /* a44 */ 00262 00263 /* Packed storage of the upper triangle of A: */ 00264 00265 /* AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ] */ 00266 00267 /* ===================================================================== */ 00268 00269 /* .. Parameters .. */ 00270 /* .. */ 00271 /* .. Local Scalars .. */ 00272 /* .. */ 00273 /* .. External Functions .. */ 00274 /* .. */ 00275 /* .. External Subroutines .. */ 00276 /* .. */ 00277 /* .. Intrinsic Functions .. */ 00278 /* .. */ 00279 /* .. Executable Statements .. */ 00280 00281 /* Parameter adjustments */ 00282 --ap; 00283 --afp; 00284 --s; 00285 b_dim1 = *ldb; 00286 b_offset = 1 + b_dim1; 00287 b -= b_offset; 00288 x_dim1 = *ldx; 00289 x_offset = 1 + x_dim1; 00290 x -= x_offset; 00291 --ferr; 00292 --berr; 00293 --work; 00294 --rwork; 00295 00296 /* Function Body */ 00297 *info = 0; 00298 nofact = lsame_(fact, "N"); 00299 equil = lsame_(fact, "E"); 00300 if (nofact || equil) { 00301 *(unsigned char *)equed = 'N'; 00302 rcequ = FALSE_; 00303 } else { 00304 rcequ = lsame_(equed, "Y"); 00305 smlnum = slamch_("Safe minimum"); 00306 bignum = 1.f / smlnum; 00307 } 00308 00309 /* Test the input parameters. */ 00310 00311 if (! nofact && ! equil && ! lsame_(fact, "F")) { 00312 *info = -1; 00313 } else if (! lsame_(uplo, "U") && ! lsame_(uplo, 00314 "L")) { 00315 *info = -2; 00316 } else if (*n < 0) { 00317 *info = -3; 00318 } else if (*nrhs < 0) { 00319 *info = -4; 00320 } else if (lsame_(fact, "F") && ! (rcequ || lsame_( 00321 equed, "N"))) { 00322 *info = -7; 00323 } else { 00324 if (rcequ) { 00325 smin = bignum; 00326 smax = 0.f; 00327 i__1 = *n; 00328 for (j = 1; j <= i__1; ++j) { 00329 /* Computing MIN */ 00330 r__1 = smin, r__2 = s[j]; 00331 smin = dmin(r__1,r__2); 00332 /* Computing MAX */ 00333 r__1 = smax, r__2 = s[j]; 00334 smax = dmax(r__1,r__2); 00335 /* L10: */ 00336 } 00337 if (smin <= 0.f) { 00338 *info = -8; 00339 } else if (*n > 0) { 00340 scond = dmax(smin,smlnum) / dmin(smax,bignum); 00341 } else { 00342 scond = 1.f; 00343 } 00344 } 00345 if (*info == 0) { 00346 if (*ldb < max(1,*n)) { 00347 *info = -10; 00348 } else if (*ldx < max(1,*n)) { 00349 *info = -12; 00350 } 00351 } 00352 } 00353 00354 if (*info != 0) { 00355 i__1 = -(*info); 00356 xerbla_("CPPSVX", &i__1); 00357 return 0; 00358 } 00359 00360 if (equil) { 00361 00362 /* Compute row and column scalings to equilibrate the matrix A. */ 00363 00364 cppequ_(uplo, n, &ap[1], &s[1], &scond, &amax, &infequ); 00365 if (infequ == 0) { 00366 00367 /* Equilibrate the matrix. */ 00368 00369 claqhp_(uplo, n, &ap[1], &s[1], &scond, &amax, equed); 00370 rcequ = lsame_(equed, "Y"); 00371 } 00372 } 00373 00374 /* Scale the right-hand side. */ 00375 00376 if (rcequ) { 00377 i__1 = *nrhs; 00378 for (j = 1; j <= i__1; ++j) { 00379 i__2 = *n; 00380 for (i__ = 1; i__ <= i__2; ++i__) { 00381 i__3 = i__ + j * b_dim1; 00382 i__4 = i__; 00383 i__5 = i__ + j * b_dim1; 00384 q__1.r = s[i__4] * b[i__5].r, q__1.i = s[i__4] * b[i__5].i; 00385 b[i__3].r = q__1.r, b[i__3].i = q__1.i; 00386 /* L20: */ 00387 } 00388 /* L30: */ 00389 } 00390 } 00391 00392 if (nofact || equil) { 00393 00394 /* Compute the Cholesky factorization A = U'*U or A = L*L'. */ 00395 00396 i__1 = *n * (*n + 1) / 2; 00397 ccopy_(&i__1, &ap[1], &c__1, &afp[1], &c__1); 00398 cpptrf_(uplo, n, &afp[1], info); 00399 00400 /* Return if INFO is non-zero. */ 00401 00402 if (*info > 0) { 00403 *rcond = 0.f; 00404 return 0; 00405 } 00406 } 00407 00408 /* Compute the norm of the matrix A. */ 00409 00410 anorm = clanhp_("I", uplo, n, &ap[1], &rwork[1]); 00411 00412 /* Compute the reciprocal of the condition number of A. */ 00413 00414 cppcon_(uplo, n, &afp[1], &anorm, rcond, &work[1], &rwork[1], info); 00415 00416 /* Compute the solution matrix X. */ 00417 00418 clacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx); 00419 cpptrs_(uplo, n, nrhs, &afp[1], &x[x_offset], ldx, info); 00420 00421 /* Use iterative refinement to improve the computed solution and */ 00422 /* compute error bounds and backward error estimates for it. */ 00423 00424 cpprfs_(uplo, n, nrhs, &ap[1], &afp[1], &b[b_offset], ldb, &x[x_offset], 00425 ldx, &ferr[1], &berr[1], &work[1], &rwork[1], info); 00426 00427 /* Transform the solution matrix X to a solution of the original */ 00428 /* system. */ 00429 00430 if (rcequ) { 00431 i__1 = *nrhs; 00432 for (j = 1; j <= i__1; ++j) { 00433 i__2 = *n; 00434 for (i__ = 1; i__ <= i__2; ++i__) { 00435 i__3 = i__ + j * x_dim1; 00436 i__4 = i__; 00437 i__5 = i__ + j * x_dim1; 00438 q__1.r = s[i__4] * x[i__5].r, q__1.i = s[i__4] * x[i__5].i; 00439 x[i__3].r = q__1.r, x[i__3].i = q__1.i; 00440 /* L40: */ 00441 } 00442 /* L50: */ 00443 } 00444 i__1 = *nrhs; 00445 for (j = 1; j <= i__1; ++j) { 00446 ferr[j] /= scond; 00447 /* L60: */ 00448 } 00449 } 00450 00451 /* Set INFO = N+1 if the matrix is singular to working precision. */ 00452 00453 if (*rcond < slamch_("Epsilon")) { 00454 *info = *n + 1; 00455 } 00456 00457 return 0; 00458 00459 /* End of CPPSVX */ 00460 00461 } /* cppsvx_ */