Data update

Task/Steffensens-method/Julia/steffensens-method.julia

@@ -0,0 +1,57 @@
+""" Aitken's extrapolation """
+function aitken(f, p0)
+    p1 = f(p0)
+    p2 = f(p1)
+    return p0 - (p1 - p0)^2 / (p2 - 2 * p1 + p0)
+end
+
+""" Steffensen's method using Aitken """
+function steffensen_aitken(f, pinit, tol, maxiter)
+    p0 = pinit
+    p = aitken(f, p0)
+    iter = 1
+    while abs(p - p0) > tol && iter < maxiter
+        p0 = p
+        p = aitken(f, p0)
+        iter += 1
+    end
+    return abs(p - p0) > tol ? NaN : p
+end
+
+""" deCasteljau function """
+function deCasteljau(c0, c1, c2, t)
+    s = 1.0 - t
+    return s * (s * c0 + t * c1) + t * (s * c1 + t * c2)
+end
+
+xConvexLeftParabola(t) = deCasteljau(2, -8, 2, t)
+yConvexRightParabola(t) = deCasteljau(1, 2, 3, t)
+implicit_equation(x, y) = 5 * x^2 + y - 5
+
+""" may return NaN on overflow """
+function f(t)
+    return t in [nothing, NaN, Inf, -Inf] ? NaN :
+       implicit_equation(xConvexLeftParabola(t), yConvexRightParabola(t)) + t
+end
+
+""" test the example """
+function test_steffensen(tol = 0.00000001, iters = 1000, stepsize = 0.1)
+    for t0 in 0:stepsize:1.1
+        print("t0 = $t0 : ")
+        t = steffensen_aitken(f, t0, tol, iters)
+        if isnan(t)
+            println("no answer")
+        else
+            x = xConvexLeftParabola(t)
+            y = yConvexRightParabola(t)
+            if abs(implicit_equation(x, y)) <= tol
+                println("intersection at ($(Float32(x)), $(Float32(y)))")
+            else
+                println("spurious solution")
+            end
+        end
+    end
+    return 0
+end
+
+test_steffensen()

Task/Steffensens-method/Raku/steffensens-method.raku

@@ -0,0 +1,48 @@
+# 20230928 Raku programming solution
+
+sub aitken($f, $p0) {
+   my $p2   = $f( my $p1 = $f($p0) );
+   my $p1m0 = $p1 - $p0;
+   return $p0 - $p1m0*$p1m0/($p2-2.0*$p1+$p0);
+}
+
+sub steffensenAitken($f, $pinit, $tol, $maxiter) {
+   my ($iter, $p) = 1, aitken($f, my $p0 = $pinit);
+   while abs($p-$p0) > $tol and $iter < $maxiter {
+      $p = aitken($f, $p0 = $p);
+      $iter++
+   }
+   return abs($p-$p0) > $tol ?? NaN !! $p
+}
+
+sub deCasteljau($c0, $c1, $c2, $t) {
+   my $s = 1.0 - $t;
+   return $s*($s*$c0 + $t*$c1) + $t*($s*$c1 + $t*$c2)
+}
+
+sub xConvexLeftParabola($t) { return deCasteljau(2.0, -8.0, 2.0, $t) }
+
+sub yConvexRightParabola($t) { return deCasteljau(1.0, 2.0, 3.0, $t) }
+
+sub implicitEquation($x, $y) { return 5.0*$x*$x + $y - 5.0 }
+
+sub f($t) {
+   implicitEquation(xConvexLeftParabola($t), yConvexRightParabola($t)) + $t
+}
+
+my $t0 = 0.0;
+for ^11 {
+   print "t0 = {$t0.fmt: '%0.1f'} : ";
+   my $t = steffensenAitken(&f, $t0, 0.00000001, 1000);
+   if $t.isNaN {
+      say "no answer";
+   } else {
+      my ($x, $y) = xConvexLeftParabola($t), yConvexRightParabola($t);
+      if abs(implicitEquation($x, $y)) <= 0.000001 {
+         printf "intersection at (%f, %f)\n", $x, $y;
+      } else {
+         say "spurious solution";
+      }
+   }
+   $t0 += 0.1;
+}