multithreaded numerical integration

acb_calc/integrate_gl_auto_deg.c

@@ -63,10 +63,28 @@ void gl_init()
 }
 
 /* Compute GL node and weight of index k for n = gl_steps[i]. Cached. */
-void
-acb_calc_gl_node(arb_t x, arb_t w, slong i, slong k, slong prec)
+typedef struct
 {
-    slong n, kk, jj, wp;
+    arb_ptr nodes;
+    arb_ptr weights;
+    slong n;
+    slong wp;
+}
+nodes_work_t;
+
+static void
+nodes_worker(slong jj, nodes_work_t * work)
+{
+    arb_hypgeom_legendre_p_ui_root(work->nodes + jj, work->weights + jj, work->n, jj, work->wp);
+}
+
+/* if k >= 0, compute the node and weight of index k */
+/* if k < 0, compute the first (n+1)/2 nodes and weights (the others are given by symmetry) */
+void
+acb_calc_gl_node(arb_ptr x, arb_ptr w, slong i, slong k, slong prec)
+{
+    slong n, kk, wp;
+    int all;
 
     if (i < 0 || i >= GL_STEPS || prec < 2)
         flint_abort();
@@ -76,16 +94,15 @@ acb_calc_gl_node(arb_t x, arb_t w, slong i, slong k, slong prec)
 
     n = gl_steps[i];
 
-    if (k < 0 || k >= n)
+    if (k >= n)
         flint_abort();
 
-    if (2 * k < n)
-        kk = k;
-    else
-        kk = n - 1 - k;
-
+    all = (k < 0);
+    
     if (gl_cache->gl_prec[i] < prec)
     {
+        nodes_work_t work;
+
         if (gl_cache->gl_prec[i] == 0)
         {
             gl_cache->gl_nodes[i] = _arb_vec_init((n + 1) / 2);
@@ -94,21 +111,86 @@ acb_calc_gl_node(arb_t x, arb_t w, slong i, slong k, slong prec)
 
         wp = FLINT_MAX(prec, gl_cache->gl_prec[i] * 2 + 30);
 
-        for (jj = 0; 2 * jj < n; jj++)
-        {
-            arb_hypgeom_legendre_p_ui_root(gl_cache->gl_nodes[i] + jj,
-                gl_cache->gl_weights[i] + jj, n, jj, wp);
-        }
+        work.nodes = gl_cache->gl_nodes[i];
+        work.weights = gl_cache->gl_weights[i];
+        work.n = n;
+        work.wp = wp;
+
+        flint_parallel_do((do_func_t) nodes_worker, &work, (n + 1) / 2, -1, FLINT_PARALLEL_STRIDED);
 
         gl_cache->gl_prec[i] = wp;
     }
 
-    if (2 * k < n)
-        arb_set_round(x, gl_cache->gl_nodes[i] + kk, prec);
+    if (all)
+    {
+        for (k = 0; k < (n + 1) / 2; k++)
+        {
+            arb_set_round(x + k, gl_cache->gl_nodes[i] + k, prec);
+            arb_set_round(w + k, gl_cache->gl_weights[i] + k, prec);
+        }
+    }
     else
-        arb_neg_round(x, gl_cache->gl_nodes[i] + kk, prec);
+    {
+        if (2 * k < n)
+            kk = k;
+        else
+            kk = n - 1 - k;
 
-    arb_set_round(w, gl_cache->gl_weights[i] + kk, prec);
+        if (2 * k < n)
+            arb_set_round(x, gl_cache->gl_nodes[i] + kk, prec);
+        else
+            arb_neg_round(x, gl_cache->gl_nodes[i] + kk, prec);
+
+        arb_set_round(w, gl_cache->gl_weights[i] + kk, prec);
+    }
+}
+
+typedef struct
+{
+    slong n;
+    slong prec;
+    arb_srcptr x;
+    arb_srcptr w;
+    acb_srcptr delta;
+    acb_srcptr mid;
+    acb_ptr v;
+    acb_calc_func_t f;
+    void *  param;
+}
+gl_work_t;
+
+static void
+gl_worker(slong k, gl_work_t * args)
+{
+    arb_t x, w;
+    acb_t t;
+    slong k2;
+
+    slong prec = args->prec;
+    slong n = args->n;
+    acb_ptr v = args->v;
+
+    arb_init(x);
+    arb_init(w);
+    acb_init(t);
+
+    if (2 * k < n)
+        k2 = k;
+    else
+        k2 = n - 1 - k;
+
+    acb_mul_arb(t, args->delta, args->x + k2, prec);
+
+    if (k2 != k)
+        acb_neg(t, t);
+
+    acb_add(t, t, args->mid, prec);
+    args->f(v + k, t, args->param, 0, prec);
+    acb_mul_arb(v + k, v + k, args->w + k2, prec);
+
+    arb_clear(x);
+    arb_clear(w);
+    acb_clear(t);
 }
 
 int
@@ -236,6 +318,7 @@ acb_calc_integrate_gl_auto_deg(acb_t res, slong * eval_count,
     /* Evaluate best found Gauss-Legendre quadrature rule. */
     if (status == ARB_CALC_SUCCESS)
     {
+        slong nt;
         arb_t x, w;
         arb_init(x);
         arb_init(w);
@@ -259,15 +342,52 @@ acb_calc_integrate_gl_auto_deg(acb_t res, slong * eval_count,
             if (gl_steps[i] == best_n)
                 break;
 
-        acb_zero(s);
+        nt = flint_get_num_threads();
 
-        for (k = 0; k < best_n; k++)
+        if (nt >= 2 && best_n >= 2)
         {
-            acb_calc_gl_node(x, w, i, k, prec);
-            acb_mul_arb(wide, delta, x, prec);
-            acb_add(wide, wide, mid, prec);
-            f(v, wide, param, 0, prec);
-            acb_addmul_arb(s, v, w, prec);
+            gl_work_t work;
+            acb_ptr v;
+            arb_ptr x, w;
+
+            v = _acb_vec_init(best_n);
+            w = _arb_vec_init((best_n + 1) / 2);
+            x = _arb_vec_init((best_n + 1) / 2);
+
+            acb_calc_gl_node(x, w, i, -1, prec);
+
+            work.n = best_n;
+            work.x = x;
+            work.w = w;
+            work.prec = prec;
+            work.delta = delta;
+            work.mid = mid;
+            work.v = v;
+            work.f = f;
+            work.param = param;
+
+            flint_parallel_do((do_func_t) gl_worker, &work, best_n, -1, FLINT_PARALLEL_STRIDED);
+
+            acb_add(s, v, v + 1, prec);
+            for (k = 2; k < best_n; k++)
+                acb_add(s, s, v + k, prec);
+
+            _acb_vec_clear(v, best_n);
+            _arb_vec_clear(x, (best_n + 1) / 2);
+            _arb_vec_clear(w, (best_n + 1) / 2);
+        }
+        else
+        {
+            acb_zero(s);
+
+            for (k = 0; k < best_n; k++)
+            {
+                acb_calc_gl_node(x, w, i, k, prec);
+                acb_mul_arb(wide, delta, x, prec);
+                acb_add(wide, wide, mid, prec);
+                f(v, wide, param, 0, prec);
+                acb_addmul_arb(s, v, w, prec);
+            }
         }
 
         eval_count[0] += best_n;

acb_calc/test/t-integrate.c

@@ -351,6 +351,8 @@ int main()
         acb_calc_integrate_opt_t opt;
         int integral;
 
+        flint_set_num_threads(1 + n_randint(state, 3));
+
         acb_init(ans);
         acb_init(res);
         acb_init(a);
@@ -712,7 +714,7 @@ int main()
     }
 
     flint_randclear(state);
-    flint_cleanup();
+    flint_cleanup_master();
     flint_printf("PASS\n");
     return EXIT_SUCCESS;
 }

doc/source/examples.rst

@@ -655,6 +655,7 @@ Invoking the program without parameters shows usage::
     -deg n     - use quadrature degree up to n
     -eval n    - limit number of function evaluations to n
     -depth n   - limit subinterval queue size to n
+    -threads n - use parallel computation with n threads
 
     Implemented integrals:
     I0 = int_0^100 sin(x) dx

examples/integrals.c

@@ -780,6 +780,7 @@ int main(int argc, char *argv[])
 {
     acb_t s, t, a, b;
     mag_t tol;
+    slong num_threads;
     slong prec, goal;
     slong N;
     ulong k;
@@ -821,7 +822,8 @@ int main(int argc, char *argv[])
         flint_printf("-verbose2  - show more information\n");
         flint_printf("-deg n     - use quadrature degree up to n\n");
         flint_printf("-eval n    - limit number of function evaluations to n\n");
-        flint_printf("-depth n   - limit subinterval queue size to n\n\n");
+        flint_printf("-depth n   - limit subinterval queue size to n\n");
+        flint_printf("-threads n - use parallel computation with n threads\n\n");
         flint_printf("Implemented integrals:\n");
         for (integral = 0; integral < NUM_INTEGRALS; integral++)
             flint_printf("I%d = %s\n", integral, descr[integral]);
@@ -835,6 +837,7 @@ int main(int argc, char *argv[])
     twice = 0;
     goal = 0;
     havetol = havegoal = 0;
+    num_threads = 1;
 
     acb_init(a);
     acb_init(b);
@@ -895,8 +898,15 @@ int main(int argc, char *argv[])
         {
             options->use_heap = 1;
         }
+        else if (!strcmp(argv[i], "-threads"))
+        {
+            num_threads = atol(argv[i+1]);
+        }
     }
 
+    if (num_threads >= 2)
+        flint_set_num_threads(num_threads);
+
     if (!havegoal)
         goal = prec;
 
@@ -1276,7 +1286,7 @@ int main(int argc, char *argv[])
     acb_clear(t);
     mag_clear(tol);
 
-    flint_cleanup();
+    flint_cleanup_master();
     return 0;
 }