void
arb_mat_pow_ui(arb_mat_t B, const arb_mat_t A, ulong exp, slong prec)
{
slong d = arb_mat_nrows(A);
if (exp <= 2 || d <= 1)
{
if (exp == 0 || d == 0)
{
arb_mat_one(B);
}
else if (d == 1)
{
arb_pow_ui(arb_mat_entry(B, 0, 0),
arb_mat_entry(A, 0, 0), exp, prec);
}
else if (exp == 1)
{
arb_mat_set(B, A);
}
else if (exp == 2)
{
arb_mat_sqr(B, A, prec);
}
}
else
{
arb_mat_t T, U;
slong i;
arb_mat_init(T, d, d);
arb_mat_set(T, A);
arb_mat_init(U, d, d);
for (i = ((slong) FLINT_BIT_COUNT(exp)) - 2; i >= 0; i--)
{
arb_mat_sqr(U, T, prec);
if (exp & (WORD(1) << i))
arb_mat_mul(T, U, A, prec);
else
arb_mat_swap(T, U);
}
arb_mat_swap(B, T);
arb_mat_clear(T);
arb_mat_clear(U);
}
}
int
arb_mat_lu(long * P, arb_mat_t LU, const arb_mat_t A, long prec)
{
arb_t d, e;
arb_ptr * a;
long i, j, m, n, r, row, col;
int result;
m = arb_mat_nrows(A);
n = arb_mat_ncols(A);
result = 1;
if (m == 0 || n == 0)
return result;
arb_mat_set(LU, A);
a = LU->rows;
row = col = 0;
for (i = 0; i < m; i++)
P[i] = i;
arb_init(d);
arb_init(e);
while (row < m && col < n)
{
r = arb_mat_find_pivot_partial(LU, row, m, col);
if (r == -1)
{
result = 0;
break;
}
else if (r != row)
arb_mat_swap_rows(LU, P, row, r);
arb_set(d, a[row] + col);
for (j = row + 1; j < m; j++)
{
arb_div(e, a[j] + col, d, prec);
arb_neg(e, e);
_arb_vec_scalar_addmul(a[j] + col,
a[row] + col, n - col, e, prec);
arb_zero(a[j] + col);
arb_neg(a[j] + row, e);
}
row++;
col++;
}
arb_clear(d);
arb_clear(e);
return result;
}
/* Calculate the likelihood, storing many intermediate calculations. */
void
evaluate_site_lhood(arb_t lhood,
arb_mat_struct *lhood_node_vectors,
arb_mat_struct *lhood_edge_vectors,
const arb_mat_struct *base_node_vectors,
const root_prior_t r, const arb_struct *equilibrium,
const arb_mat_struct *transition_matrices,
const csr_graph_struct *g,
const int *preorder, int node_count, slong prec)
{
int u, a, b, idx;
int start, stop;
arb_mat_struct *nmat, *nmatb, *emat;
const arb_mat_struct *tmat;
for (u = 0; u < node_count; u++)
{
a = preorder[node_count - 1 - u];
nmat = lhood_node_vectors + a;
start = g->indptr[a];
stop = g->indptr[a+1];
/* initialize the state vector for node a */
arb_mat_set(nmat, base_node_vectors + a);
/* create all of the state vectors on edges outgoing from this node */
for (idx = start; idx < stop; idx++)
{
b = g->indices[idx];
/*
* At this point (a, b) is an edge from node a to node b
* in a post-order traversal of edges of the tree.
*/
tmat = transition_matrices + idx;
nmatb = lhood_node_vectors + b;
/*
* Update vectors on edges if requested,
* otherwise avoid using temporary vectors on edges.
* In either case, lhood node vectors are updated.
*/
if (lhood_edge_vectors)
{
emat = lhood_edge_vectors + idx;
_arb_mat_mul_stochastic(emat, tmat, nmatb, prec);
arb_mat_mul_entrywise(nmat, nmat, emat, prec);
}
else
{
_prune_update_prob(nmat, nmat, tmat, nmatb, prec);
}
}
}
/* Report the sum of state entries associated with the root. */
int root_node_index = preorder[0];
nmat = lhood_node_vectors + root_node_index;
root_prior_expectation(lhood, r, nmat, equilibrium, prec);
}
void
cross_site_ws_update_with_edge_rates(cross_site_ws_t w,
const model_and_data_t m, const arb_struct *edge_rates, slong prec)
{
w->prec = prec;
/* update rate mixture */
rate_mixture_summarize(
w->rate_mix_prior, w->rate_mix_rates, w->rate_mix_expect,
m->rate_mixture, prec);
/* update the equilibrium if necessary, ignoring diagonal entries */
if (model_and_data_uses_equilibrium(m))
{
_arb_vec_rate_matrix_equilibrium(w->equilibrium, m->mat, prec);
}
/* update the unscaled rate matrix, and zero the diagonal */
arb_mat_set(w->rate_matrix, m->mat);
_arb_mat_zero_diagonal(w->rate_matrix);
/* update the rate divisor, optionally using the equilibrium */
_update_rate_divisor(w, m, prec);
/*
* Scale the rate matrix according to the rate divisor.
* Each rate matrix associated with a specific rate category
* and branch will need to be further scaled.
*/
arb_mat_scalar_div_arb(w->rate_matrix,
w->rate_matrix, w->rate_divisor, prec);
/* update the diagonal of the rate matrix */
_arb_update_rate_matrix_diagonal(w->rate_matrix, prec);
/* optionally set edge rates */
if (edge_rates)
{
_arb_vec_set(w->edge_rates, edge_rates, w->edge_count);
}
/* update the transition probability matrices */
_update_transition_matrices(w, prec);
}
void
arb_mat_solve_cho_precomp(arb_mat_t X,
const arb_mat_t L, const arb_mat_t B, slong prec)
{
slong i, j, c, n, m;
n = arb_mat_nrows(X);
m = arb_mat_ncols(X);
arb_mat_set(X, B);
for (c = 0; c < m; c++)
{
/* solve Ly = b */
for (i = 0; i < n; i++)
{
for (j = 0; j < i; j++)
{
arb_submul(arb_mat_entry(X, i, c),
arb_mat_entry(L, i, j), arb_mat_entry(X, j, c), prec);
}
arb_div(arb_mat_entry(X, i, c), arb_mat_entry(X, i, c),
arb_mat_entry(L, i, i), prec);
}
/* solve Ux = y */
for (i = n - 1; i >= 0; i--)
{
for (j = i + 1; j < n; j++)
{
arb_submul(arb_mat_entry(X, i, c),
arb_mat_entry(L, j, i), arb_mat_entry(X, j, c), prec);
}
arb_div(arb_mat_entry(X, i, c), arb_mat_entry(X, i, c),
arb_mat_entry(L, i, i), prec);
}
}
}
static int
_spd_inv(arb_mat_t X, const arb_mat_t A, slong prec)
{
slong n;
arb_mat_t L;
int result;
n = arb_mat_nrows(A);
arb_mat_init(L, n, n);
arb_mat_set(L, A);
if (_arb_mat_ldl_inplace(L, prec))
{
arb_mat_inv_ldl_precomp(X, L, prec);
result = 1;
}
else
{
result = 0;
}
arb_mat_clear(L);
return result;
}
int arb_mat_jacobi(arb_mat_t D, arb_mat_t P, const arb_mat_t A, slong prec) {
//
// Given a d x d real symmetric matrix A, compute an orthogonal matrix
// P and a diagonal D such that A = P D P^t = P D P^(-1).
//
// D should have already been initialized as a d x 1 matrix, and Pp
// should have already been initialized as a d x d matrix.
//
// If the eigenvalues can be certified as unique, then a nonzero int is
// returned, and the eigenvectors should have reasonable error bounds. If
// the eigenvalues cannot be certified as unique, then some of the
// eigenvectors will have infinite error radius.
#define B(i,j) arb_mat_entry(B, i, j)
#define D(i) arb_mat_entry(D, i, 0)
#define P(i,j) arb_mat_entry(P, i, j)
int dim = arb_mat_nrows(A);
if(dim == 1) {
arb_mat_set(D, A);
arb_mat_one(P);
return 0;
}
arb_mat_t B;
arb_mat_init(B, dim, dim);
arf_t * B1 = (arf_t*)malloc(dim * sizeof(arf_t));
arf_t * B2 = (arf_t*)malloc(dim * sizeof(arf_t));
arf_t * row_max = (arf_t*)malloc((dim - 1) * sizeof(arf_t));
int * row_max_indices = (int*)malloc((dim - 1) * sizeof(int));
for(int k = 0; k < dim; k++) {
arf_init(B1[k]);
arf_init(B2[k]);
}
for(int k = 0; k < dim - 1; k++) {
arf_init(row_max[k]);
}
arf_t x1, x2;
arf_init(x1);
arf_init(x2);
arf_t Gii, Gij, Gji, Gjj;
arf_init(Gii);
arf_init(Gij);
arf_init(Gji);
arf_init(Gjj);
arb_mat_set(B, A);
arb_mat_one(P);
for(int i = 0; i < dim - 1; i++) {
for(int j = i + 1; j < dim; j++) {
arf_abs(x1, arb_midref(B(i,j)));
if(arf_cmp(row_max[i], x1) < 0) {
arf_set(row_max[i], x1);
row_max_indices[i] = j;
}
}
}
int finished = 0;
while(!finished) {
arf_zero(x1);
int i = 0;
int j = 0;
for(int k = 0; k < dim - 1; k++) {
if(arf_cmp(x1, row_max[k]) < 0) {
arf_set(x1, row_max[k]);
i = k;
}
}
j = row_max_indices[i];
slong bound = arf_abs_bound_lt_2exp_si(x1);
if(bound < -prec * .9) {
finished = 1;
break;
}
else {
//printf("%ld\n", arf_abs_bound_lt_2exp_si(x1));
//arb_mat_printd(B, 10);
//printf("\n");
}
arf_twobytwo_diag(Gii, Gij, arb_midref(B(i,i)), arb_midref(B(i,j)), arb_midref(B(j,j)), 2*prec);
arf_neg(Gji, Gij);
arf_set(Gjj, Gii);
//printf("%d %d\n", i, j);
//arf_printd(Gii, 100);
//printf(" ");
//arf_printd(Gij, 100);
//printf("\n");
if(arf_is_zero(Gij)) { // If this happens, we're
finished = 1; // not going to do any better
break; // without increasing the precision.
}
for(int k = 0; k < dim; k++) {
arf_mul(B1[k], Gii, arb_midref(B(i,k)), prec, ARF_RND_NEAR);
arf_addmul(B1[k], Gji, arb_midref(B(j,k)), prec, ARF_RND_NEAR);
arf_mul(B2[k], Gij, arb_midref(B(i,k)), prec, ARF_RND_NEAR);
arf_addmul(B2[k], Gjj, arb_midref(B(j,k)), prec, ARF_RND_NEAR);
}
for(int k = 0; k < dim; k++) {
arf_set(arb_midref(B(i,k)), B1[k]);
arf_set(arb_midref(B(j,k)), B2[k]);
}
for(int k = 0; k < dim; k++) {
arf_mul(B1[k], Gii, arb_midref(B(k,i)), prec, ARF_RND_NEAR);
arf_addmul(B1[k], Gji, arb_midref(B(k,j)), prec, ARF_RND_NEAR);
arf_mul(B2[k], Gij, arb_midref(B(k,i)), prec, ARF_RND_NEAR);
arf_addmul(B2[k], Gjj, arb_midref(B(k,j)), prec, ARF_RND_NEAR);
}
for(int k = 0; k < dim; k++) {
arf_set(arb_midref(B(k,i)), B1[k]);
arf_set(arb_midref(B(k,j)), B2[k]);
}
for(int k = 0; k < dim; k++) {
arf_mul(B1[k], Gii, arb_midref(P(k,i)), prec, ARF_RND_NEAR);
arf_addmul(B1[k], Gji, arb_midref(P(k,j)), prec, ARF_RND_NEAR);
arf_mul(B2[k], Gij, arb_midref(P(k,i)), prec, ARF_RND_NEAR);
arf_addmul(B2[k], Gjj, arb_midref(P(k,j)), prec, ARF_RND_NEAR);
}
for(int k = 0; k < dim; k++) {
arf_set(arb_midref(P(k,i)), B1[k]);
arf_set(arb_midref(P(k,j)), B2[k]);
}
if(i < dim - 1)
arf_set_ui(row_max[i], 0);
if(j < dim - 1)
arf_set_ui(row_max[j], 0);
// Update the max in any row where the maximum
// was in a column that changed.
for(int k = 0; k < dim - 1; k++) {
if(row_max_indices[k] == j || row_max_indices[k] == i) {
arf_abs(row_max[k], arb_midref(B(k,k+1)));
row_max_indices[k] = k+1;
for(int l = k+2; l < dim; l++) {
arf_abs(x1, arb_midref(B(k,l)));
if(arf_cmp(row_max[k], x1) < 0) {
arf_set(row_max[k], x1);
row_max_indices[k] = l;
}
}
}
}
// Update the max in the ith row.
for(int k = i + 1; k < dim; k++) {
arf_abs(x1, arb_midref(B(i, k)));
if(arf_cmp(row_max[i], x1) < 0) {
arf_set(row_max[i], x1);
row_max_indices[i] = k;
}
}
// Update the max in the jth row.
for(int k = j + 1; k < dim; k++) {
arf_abs(x1, arb_midref(B(j, k)));
if(arf_cmp(row_max[j], x1) < 0) {
arf_set(row_max[j], x1);
row_max_indices[j] = k;
}
}
// Go through column i to see if any of
// the new entries are larger than the
// max of their row.
for(int k = 0; k < i; k++) {
if(k == dim) continue;
arf_abs(x1, arb_midref(B(k, i)));
if(arf_cmp(row_max[k], x1) < 0) {
arf_set(row_max[k], x1);
row_max_indices[k] = i;
}
}
// And then column j.
for(int k = 0; k < j; k++) {
if(k == dim) continue;
arf_abs(x1, arb_midref(B(k, j)));
if(arf_cmp(row_max[k], x1) < 0) {
arf_set(row_max[k], x1);
row_max_indices[k] = j;
}
}
}
for(int k = 0; k < dim; k++) {
arb_set(D(k), B(k,k));
arb_set_exact(D(k));
}
// At this point we've done that diagonalization and all that remains is
// to certify the correctness and compute error bounds.
arb_mat_t e;
arb_t error_norms[dim];
for(int k = 0; k < dim; k++) arb_init(error_norms[k]);
arb_mat_init(e, dim, 1);
arb_t z1, z2;
arb_init(z1);
arb_init(z2);
for(int j = 0; j < dim; j++) {
arb_mat_set(B, A);
for(int k = 0; k < dim; k++) {
arb_sub(B(k, k), B(k, k), D(j), prec);
}
for(int k = 0; k < dim; k++) {
arb_set(arb_mat_entry(e, k, 0), P(k, j));
}
arb_mat_L2norm(z2, e, prec);
arb_mat_mul(e, B, e, prec);
arb_mat_L2norm(error_norms[j], e, prec);
arb_div(z2, error_norms[j], z2, prec); // and now z1 is an upper bound for the
// error in the eigenvalue
arb_add_error(D(j), z2);
}
int unique_eigenvalues = 1;
for(int j = 0; j < dim; j++) {
if(j == 0) {
arb_sub(z1, D(j), D(1), prec);
}
else {
arb_sub(z1, D(j), D(0), prec);
}
arb_get_abs_lbound_arf(x1, z1, prec);
for(int k = 1; k < dim; k++) {
if(k == j) continue;
arb_sub(z1, D(j), D(k), prec);
arb_get_abs_lbound_arf(x2, z1, prec);
if(arf_cmp(x2, x1) < 0) {
arf_set(x1, x2);
}
}
if(arf_is_zero(x1)) {
unique_eigenvalues = 0;
}
arb_div_arf(z1, error_norms[j], x1, prec);
for(int k = 0; k < dim; k++) {
arb_add_error(P(k, j), z1);
}
}
arb_mat_clear(e);
arb_clear(z1);
arb_clear(z2);
for(int k = 0; k < dim; k++) arb_clear(error_norms[k]);
arf_clear(x1);
arf_clear(x2);
arb_mat_clear(B);
for(int k = 0; k < dim; k++) {
arf_clear(B1[k]);
arf_clear(B2[k]);
}
for(int k = 0; k < dim - 1; k++) {
arf_clear(row_max[k]);
}
arf_clear(Gii);
arf_clear(Gij);
arf_clear(Gji);
arf_clear(Gjj);
free(B1);
free(B2);
free(row_max);
free(row_max_indices);
if(unique_eigenvalues) return 0;
else return 1;
#undef B
#undef D
#undef P
}
/* evaluates the truncated Taylor series (assumes no aliasing) */
void
_arb_mat_exp_taylor(arb_mat_t S, const arb_mat_t A, slong N, slong prec)
{
if (N == 1)
{
arb_mat_one(S);
}
else if (N == 2)
{
arb_mat_one(S);
arb_mat_add(S, S, A, prec);
}
else if (N == 3)
{
arb_mat_t T;
arb_mat_init(T, arb_mat_nrows(A), arb_mat_nrows(A));
arb_mat_mul(T, A, A, prec);
arb_mat_scalar_mul_2exp_si(T, T, -1);
arb_mat_add(S, A, T, prec);
arb_mat_one(T);
arb_mat_add(S, S, T, prec);
arb_mat_clear(T);
}
else
{
slong i, lo, hi, m, w, dim;
arb_mat_struct * pows;
arb_mat_t T, U;
fmpz_t c, f;
dim = arb_mat_nrows(A);
m = n_sqrt(N);
w = (N + m - 1) / m;
fmpz_init(c);
fmpz_init(f);
pows = flint_malloc(sizeof(arb_mat_t) * (m + 1));
arb_mat_init(T, dim, dim);
arb_mat_init(U, dim, dim);
for (i = 0; i <= m; i++)
{
arb_mat_init(pows + i, dim, dim);
if (i == 0)
arb_mat_one(pows + i);
else if (i == 1)
arb_mat_set(pows + i, A);
else
arb_mat_mul(pows + i, pows + i - 1, A, prec);
}
arb_mat_zero(S);
fmpz_one(f);
for (i = w - 1; i >= 0; i--)
{
lo = i * m;
hi = FLINT_MIN(N - 1, lo + m - 1);
arb_mat_zero(T);
fmpz_one(c);
while (hi >= lo)
{
arb_mat_scalar_addmul_fmpz(T, pows + hi - lo, c, prec);
if (hi != 0)
fmpz_mul_ui(c, c, hi);
hi--;
}
arb_mat_mul(U, pows + m, S, prec);
arb_mat_scalar_mul_fmpz(S, T, f, prec);
arb_mat_add(S, S, U, prec);
fmpz_mul(f, f, c);
}
arb_mat_scalar_div_fmpz(S, S, f, prec);
fmpz_clear(c);
fmpz_clear(f);
for (i = 0; i <= m; i++)
arb_mat_clear(pows + i);
flint_free(pows);
arb_mat_clear(T);
arb_mat_clear(U);
}
}
int main()
{
slong iter;
flint_rand_t state;
flint_printf("mul_entrywise....");
fflush(stdout);
flint_randinit(state);
for (iter = 0; iter < 10000 * arb_test_multiplier(); iter++)
{
slong m, n, qbits1, qbits2, rbits1, rbits2, rbits3;
fmpq_mat_t A, B, C;
arb_mat_t a, b, c, d;
qbits1 = 2 + n_randint(state, 200);
qbits2 = 2 + n_randint(state, 200);
rbits1 = 2 + n_randint(state, 200);
rbits2 = 2 + n_randint(state, 200);
rbits3 = 2 + n_randint(state, 200);
m = n_randint(state, 10);
n = n_randint(state, 10);
fmpq_mat_init(A, m, n);
fmpq_mat_init(B, m, n);
fmpq_mat_init(C, m, n);
arb_mat_init(a, m, n);
arb_mat_init(b, m, n);
arb_mat_init(c, m, n);
arb_mat_init(d, m, n);
fmpq_mat_randtest(A, state, qbits1);
fmpq_mat_randtest(B, state, qbits2);
_fmpq_mat_mul_entrywise(C, A, B);
arb_mat_set_fmpq_mat(a, A, rbits1);
arb_mat_set_fmpq_mat(b, B, rbits2);
arb_mat_mul_entrywise(c, a, b, rbits3);
if (!arb_mat_contains_fmpq_mat(c, C))
{
flint_printf("FAIL\n\n");
flint_printf("m = %wd, n = %wd, bits3 = %wd\n", m, n, rbits3);
flint_printf("A = "); fmpq_mat_print(A); flint_printf("\n\n");
flint_printf("B = "); fmpq_mat_print(B); flint_printf("\n\n");
flint_printf("C = "); fmpq_mat_print(C); flint_printf("\n\n");
flint_printf("a = "); arb_mat_printd(a, 15); flint_printf("\n\n");
flint_printf("b = "); arb_mat_printd(b, 15); flint_printf("\n\n");
flint_printf("c = "); arb_mat_printd(c, 15); flint_printf("\n\n");
abort();
}
/* test aliasing with a */
if (arb_mat_nrows(a) == arb_mat_nrows(c) &&
arb_mat_ncols(a) == arb_mat_ncols(c))
{
arb_mat_set(d, a);
arb_mat_mul_entrywise(d, d, b, rbits3);
if (!arb_mat_equal(d, c))
{
flint_printf("FAIL (aliasing 1)\n\n");
abort();
}
}
/* test aliasing with b */
if (arb_mat_nrows(b) == arb_mat_nrows(c) &&
arb_mat_ncols(b) == arb_mat_ncols(c))
{
arb_mat_set(d, b);
arb_mat_mul_entrywise(d, a, d, rbits3);
if (!arb_mat_equal(d, c))
{
flint_printf("FAIL (aliasing 2)\n\n");
abort();
}
}
fmpq_mat_clear(A);
fmpq_mat_clear(B);
fmpq_mat_clear(C);
arb_mat_clear(a);
arb_mat_clear(b);
arb_mat_clear(c);
arb_mat_clear(d);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}