/* nag_real_sparse_eigensystem_monit (f12aec) Example Program. * * Copyright 2005 Numerical Algorithms Group. * * Mark 8, 2005. */ #include #include #include #include #include #include #include static void mv(Integer, double *, double *); static void av(Integer, double *, double *); static int ytax(Integer, double *, double *, double *); static int ytmx(Integer, double *, double *, double *); static void my_zgttrf(Integer, Complex *, Complex *, Complex *, Complex *, Integer *, Integer *); static void my_zgttrs(Integer, Complex *, Complex *, Complex *, Complex *, Integer *, Complex *); int main(void) { /* Constants */ Integer licomm = 140, imon = 1; /* Scalars */ Complex c1, c2, c3, eigv, num, den; double estnrm, deni, denr, i2, numi, numr, r2; double sigmai, sigmar; Integer exit_status, info, irevcm, j, k, lcomm, n; Integer nconv, ncv, nev, niter, nshift; /* Nag types */ Nag_Boolean first; NagError fail; /* Arrays */ Complex *cdd = 0, *cdl = 0, *cdu = 0, *cdu2 = 0, *ctemp = 0; double *comm = 0, *eigvr = 0, *eigvi = 0, *eigest = 0; double *resid = 0, *v = 0; Integer *icomm = 0, *ipiv = 0; /* Pointers */ double *mx = 0, *x = 0, *y = 0; exit_status = 0; INIT_FAIL(fail); printf("nag_real_sparse_eigensystem_monit (f12aec) Example Program " "Results\n"); /* Skip heading in data file */ scanf("%*[^\n] "); /* Read problem parameter values from data file. */ scanf("%ld%ld%ld%lf%lf%*[^\n] ", &n, &nev, &ncv, &sigmar, &sigmai); /* Allocate memory */ lcomm = 3*n + 3*ncv*ncv + 6*ncv + 60; if (!(cdd = NAG_ALLOC(n, Complex)) || !(cdl = NAG_ALLOC(n, Complex)) || !(cdu = NAG_ALLOC(n, Complex)) || !(cdu2 = NAG_ALLOC(n, Complex)) || !(ctemp = NAG_ALLOC(n, Complex)) || !(comm = NAG_ALLOC(lcomm, double)) || !(eigvr = NAG_ALLOC(ncv, double)) || !(eigvi = NAG_ALLOC(ncv, double)) || !(eigest = NAG_ALLOC(ncv, double)) || !(resid = NAG_ALLOC(n, double)) || !(v = NAG_ALLOC(n * ncv, double)) || !(icomm = NAG_ALLOC(licomm, Integer)) || !(ipiv = NAG_ALLOC(n, Integer))) { printf("Allocation failure\n"); exit_status = -1; goto END; } /* Initialise communication arrays for problem using nag_real_sparse_eigensystem_init (f12aac). */ nag_real_sparse_eigensystem_init(n, nev, ncv, icomm, licomm, comm, lcomm, &fail); if (fail.code != NE_NOERROR) { printf( "Error from nag_real_sparse_eigensystem_init (f12aac).\n%s\n", fail.message); exit_status = 1; goto END; } /* Select the required spectrum using nag_real_sparse_eigensystem_option (f12adc). */ nag_real_sparse_eigensystem_option("SHIFTED REAL", icomm, comm, &fail); /* Select the problem type using nag_real_sparse_eigensystem_option (f12adc). */ nag_real_sparse_eigensystem_option("GENERALIZED", icomm, comm, &fail); /* Solve A*x = lambda*B*x in shift-invert mode. */ /* The shift, sigma, is a complex number (sigmar, sigmai). */ /* OP = Real_Part{inv[A-(sigmar,sigmai)*M]*M and B = M. */ c1 = nag_complex(-2. - sigmar, -sigmai); c2 = nag_complex(2. - sigmar * 4., sigmai * -4.); c3 = nag_complex(3. - sigmar, -sigmai); for (j = 0; j <= n - 2; ++j) { cdl[j] = c1; cdd[j] = c2; cdu[j] = c3; } cdd[n-1] = c2; my_zgttrf(n, cdl, cdd, cdu, cdu2, ipiv, &info); irevcm = 0; REVCOMLOOP: /* repeated calls to reverse communication routine nag_real_sparse_eigensystem_iter (f12abc). */ nag_real_sparse_eigensystem_iter(&irevcm, resid, v, &x, &y, &mx, &nshift, comm, icomm, &fail); if (irevcm != 5) { if (irevcm == -1) { /* Perform x <--- OP*x = inv[A-SIGMA*M]*M*x */ mv(n, x, y); for (j = 0; j <= n-1; ++j) { ctemp[j].re = y[j], ctemp[j].im = 0.; } my_zgttrs(n, cdl, cdd, cdu, cdu2, ipiv, ctemp); for (j = 0; j <= n-1; ++j) { y[j] = ctemp[j].re; } } else if (irevcm == 1) { /* Perform x <--- OP*x = inv[A-SIGMA*M]*M*x, */ /* M*X stored in MX. */ for (j = 0; j <= n-1; ++j) { ctemp[j].re = mx[j], ctemp[j].im = 0.; } my_zgttrs(n, cdl, cdd, cdu, cdu2, ipiv, ctemp); for (j = 0; j <= n-1; ++j) { y[j] = ctemp[j].re; } } else if (irevcm == 2) { /* Perform y <--- M*x */ mv(n, x, y); } else if (irevcm == 4 && imon == 1) { /* If imon=1, get monitoring information using nag_real_sparse_eigensystem_monit (f12aec). */ nag_real_sparse_eigensystem_monit(&niter, &nconv, eigvr, eigvi, eigest, icomm, comm); /* Compute 2-norm of Ritz estimates using nag_dge_norm (f16rac).*/ nag_dge_norm(Nag_ColMajor, Nag_FrobeniusNorm, nev, 1, eigest, nev, &estnrm, &fail); printf("Iteration %3ld, ", niter); printf(" No. converged = %3ld,", nconv); printf(" norm of estimates = %17.8e\n", estnrm); } goto REVCOMLOOP; } if (fail.code == NE_NOERROR) { /* Post-Process using nag_real_sparse_eigensystem_sol (f12acc) to compute eigenvalues/vectors. */ nag_real_sparse_eigensystem_sol(&nconv, eigvr, eigvi, v, sigmar, sigmai, resid, v, comm, icomm, &fail); first = Nag_TRUE; k = 0; for (j = 0; j <= nconv-1; ++j) { /* Use Rayleigh Quotient to recover eigenvalues of the */ /* original problem. */ if (eigvi[j] == 0.) { /* Ritz value is real. */ /* Numerator = Vj . AVj where Vj is jth Ritz vector */ if (ytax(n, &v[k], &v[k], &numr)) { goto END; } /* Denominator = Vj . MVj */ if (ytmx(n, &v[k], &v[k], &denr)) { goto END; } eigvr[j] = numr / denr; } else if (first) { /* Ritz value is complex: (x,y). */ /* Compute x'(Ax) and y'(Ax). */ if (ytax(n, &v[k], &v[k], &numr)) { goto END; } if (ytax(n, &v[k], &v[k+n], &numi)) { goto END; } /* Compute y'(Ay) and x'(Ay). */ if (ytax(n, &v[k+n], &v[k+n], &r2)) { goto END; } if (ytax(n, &v[k+n], &v[k], &i2)) { goto END; } numr += r2; numi = i2 - numi; /* Assign to Complex type using nag_complex (a02bac). */ num = nag_complex(numr, numi); /* Compute x'(Mx) and y'(Mx). */ if (ytmx(n, &v[k], &v[k], &denr)) { goto END; } if (ytmx(n, &v[k], &v[k+n], &deni)) { goto END; } /* Compute y'(Ay) and x'(Ay). */ if (ytmx(n, &v[k+n], &v[k+n], &r2)) { goto END; } if (ytmx(n, &v[k+n], &v[k], &i2)) { goto END; } denr += r2; deni = i2 - deni; /* Assign to Complex type using nag_complex (a02bac). */ den = nag_complex(denr, deni); /* eigv = x'(Ax)/x'(Mx) */ /* Compute Complex division using nag_complex_divide (a02cdc). */ eigv = nag_complex_divide(num, den); eigvr[j] = eigv.re; eigvi[j] = eigv.im; first = Nag_FALSE; } else { /* Second of complex conjugate pair. */ eigvr[j] = eigvr[j-1]; eigvi[j] = -eigvi[j-1]; first = Nag_TRUE; } k = k + n; } /* Print computed eigenvalues. */ printf("\n The %4ld generalized Ritz values closest", nconv); printf(" to ( %8.4f , %8.4f ) are:\n\n", sigmar, sigmai); for (j = 0; j <= nconv-1; ++j) { printf("%8ld%5s( %7.4f, %7.4f )\n", j+1, "", eigvr[j], eigvi[j]); } } else { printf( " Error from nag_real_sparse_eigensystem_iter (f12abc).\n%s\n", fail.message); exit_status = 1; goto END; } END: if (cdd) NAG_FREE(cdd); if (cdl) NAG_FREE(cdl); if (cdu) NAG_FREE(cdu); if (cdu2) NAG_FREE(cdu2); if (ctemp) NAG_FREE(ctemp); if (comm) NAG_FREE(comm); if (eigvr) NAG_FREE(eigvr); if (eigvi) NAG_FREE(eigvi); if (eigest) NAG_FREE(eigest); if (resid) NAG_FREE(resid); if (v) NAG_FREE(v); if (icomm) NAG_FREE(icomm); if (ipiv) NAG_FREE(ipiv); return exit_status; } static void mv(Integer n, double *v, double *y) { /* Compute the matrix vector multiplication y<---M*x, */ /* where M is mass matrix formed by using piecewise linear elements */ /* on [0,1]. */ /* Scalars */ Integer j; /* Function Body */ y[0] = v[0] * 4. + v[1]; for (j = 1; j <= n - 2; ++j) { y[j] = v[j-1] + v[j] * 4. + v[j+1]; } y[n-1] = v[n-2] + v[n-1] * 4.; return; } /* mv */ static void av(Integer n, double *v, double *w) { /* Scalars */ Integer j; /* Function Body */ w[0] = v[0] * 2. + v[1] * 3.; for (j = 1; j <= n - 2; ++j) { w[j] = v[j-1] * -2. + v[j] * 2. + v[j+1] * 3.; } w[n-1] = v[n-2] * -2. + v[n-1] * 2.; return; } /* av */ static int ytax(Integer n, double x[], double y[], double *r) { /* Given the vectors x and y, Performs the operation */ /* y'Ax and returns the scalar value. */ /* Scalars */ Integer exit_status, j; /* Arrays */ double *ax = 0; /* Function Body */ exit_status = 0; /* Allocate memory */ if (!(ax = NAG_ALLOC(n, double))) { printf("Allocation failure\n"); exit_status = -1; goto YTAXEND; } av(n, x, ax); *r = 0.0; for (j = 0; j <= n - 1; ++j) { *r = *r + y[j] * ax[j]; } YTAXEND: if (ax) NAG_FREE(ax); return exit_status; } /* ytax */ static int ytmx(Integer n, double x[], double y[], double *r) { /* Given the vectors x and y, Performs the operation */ /* y'Mx and returns the scalar value. */ /* Scalars */ Integer exit_status, j; /* Arrays */ double *mx = 0; /* Function Body */ exit_status = 0; /* Allocate memory */ if (!(mx = NAG_ALLOC(n, double))) { printf("Allocation failure\n"); exit_status = -1; goto YTMXEND; } mv(n, x, mx); *r = 0.0; for (j = 0; j <= n - 1; ++j) { *r = *r + y[j] * mx[j]; } YTMXEND: if (mx) NAG_FREE(mx); return exit_status; } /* ytmx */ static void my_zgttrf(Integer n, Complex dl[], Complex d[], Complex du[], Complex du2[], Integer ipiv[], Integer *info) { /* A simple C version of the Lapack routine zgttrf with argument checking removed */ /* Scalars */ Complex temp, fact, z1; Integer i; /* Function Body */ *info = 0; for (i = 0; i < n; ++i) { ipiv[i] = i; } for (i = 0; i < n - 2; ++i) { du2[i] = nag_complex(0.0, 0.0); } for (i = 0; i < n - 2; ++i) { if (fabs(d[i].re)+fabs(d[i].im) >= fabs(dl[i].re)+fabs(dl[i].im)) { /* No row interchange required, eliminate dl[i]. */ if (fabs(d[i].re)+fabs(d[i].im) != 0.0) { /* Compute Complex division using nag_complex_divide (a02cdc). */ fact = nag_complex_divide(dl[i], d[i]); dl[i] = fact; /* Compute Complex multiply using nag_complex_multiply (a02ccc). */ fact = nag_complex_multiply(fact, du[i]); /* Compute Complex subtraction using nag_complex_subtract (a02cbc). */ d[i+1] = nag_complex_subtract(d[i+1], fact); } } else { /* Interchange rows I and I+1, eliminate dl[I] */ /* Compute Complex division using nag_complex_divide (a02cdc). */ fact = nag_complex_divide(d[i], dl[i]); d[i] = dl[i]; dl[i] = fact; temp = du[i]; du[i] = d[i+1]; /* Compute Complex multiply using nag_complex_multiply (a02ccc). */ z1 = nag_complex_multiply(fact, d[i+1]); /* Compute Complex subtraction using nag_complex_subtract (a02cbc). */ d[i+1] = nag_complex_subtract(temp, z1); du2[i] = du[i+1]; /* Compute Complex multiply using nag_complex_multiply (a02ccc). */ du[i+1] = nag_complex_multiply(fact, du[i+1]); /* Perform Complex negation using nag_complex_negate (a02cec). */ du[i+1] = nag_complex_negate(du[i+1]); ipiv[i] = i + 1; } } if (n > 1) { i = n - 2; if (fabs(d[i].re)+fabs(d[i].im) >= fabs(dl[i].re)+fabs(dl[i].im)) { if (fabs(d[i].re)+fabs(d[i].im) != 0.0) { /* Compute Complex division using nag_complex_divide (a02cdc). */ fact = nag_complex_divide(dl[i], d[i]); dl[i] = fact; /* Compute Complex multiply using nag_complex_multiply (a02ccc). */ fact = nag_complex_multiply(fact, du[i]); /* Compute Complex subtraction using nag_complex_subtract (a02cbc). */ d[i+1] = nag_complex_subtract(d[i+1], fact); } } else { /* Compute Complex division using nag_complex_divide (a02cdc). */ fact = nag_complex_divide(d[i], dl[i]); d[i] = dl[i]; dl[i] = fact; temp = du[i]; du[i] = d[i+1]; /* Compute Complex multiply using nag_complex_multiply (a02ccc). */ z1 = nag_complex_multiply(fact, d[i+1]); /* Compute Complex subtraction using nag_complex_subtract (a02cbc). */ d[i+1] = nag_complex_subtract(temp, z1); ipiv[i] = i + 1; } } /* Check for a zero on the diagonal of U. */ for (i = 0; i < n; ++i) { if (fabs(d[i].re)+fabs(d[i].im) == 0.0) { *info = i; goto END; } } END: return; } static void my_zgttrs(Integer n, Complex dl[], Complex d[], Complex du[], Complex du2[], Integer ipiv[], Complex b[]) { /* A simple C version of the Lapack routine zgttrs with argument checking removed, the number of right-hand-sides=1, Trans='N' */ /* Scalars */ Complex temp, z1; Integer i; /* Solve L*x = b. */ for (i = 0; i < n - 1; ++i) { if (ipiv[i] == i) { /* b[i+1] = b[i+1] - dl[i]*b[i] */ /* Compute Complex multiply using nag_complex_multiply (a02ccc). */ temp = nag_complex_multiply(dl[i], b[i]); /* Compute Complex subtraction using nag_complex_subtract (a02cbc). */ b[i+1] = nag_complex_subtract(b[i+1], temp); } else { temp = b[i]; b[i] = b[i+1]; /* Compute Complex multiply using nag_complex_multiply (a02ccc). */ z1 = nag_complex_multiply(dl[i], b[i]); /* Compute Complex subtraction using nag_complex_subtract (a02cbc). */ b[i+1] = nag_complex_subtract(temp, z1); } } /* Solve U*x = b. */ /* Compute Complex division using nag_complex_divide (a02cdc). */ b[n-1] = nag_complex_divide(b[n-1], d[n-1]); if (n > 1) { /* Compute Complex multiply using nag_complex_multiply (a02ccc). */ temp = nag_complex_multiply(du[n-2], b[n-1]); /* Compute Complex subtraction using nag_complex_subtract (a02cbc). */ z1 = nag_complex_subtract(b[n-2], temp); /* Compute Complex division using nag_complex_divide (a02cdc). */ b[n-2] = nag_complex_divide(z1, d[n-2]); } for (i = n - 3; i >= 0; --i) { /* b[i] = (b[i]-du[i]*b[i+1]-du2[i]*b[i+2])/d[i]; */ /* Compute Complex multiply using nag_complex_multiply (a02ccc). */ temp = nag_complex_multiply(du[i], b[i+1]); z1 = nag_complex_multiply(du2[i], b[i+2]); /* Compute Complex addition using nag_complex_add (a02cac). */ temp = nag_complex_add(temp, z1); /* Compute Complex subtraction using nag_complex_subtract (a02cbc). */ z1 = nag_complex_subtract(b[i], temp); /* Compute Complex division using nag_complex_divide (a02cdc). */ b[i] = nag_complex_divide(z1, d[i]); } return; }