Changes between Version 8 and Version 9 of research/trig Tweet

Timestamp:

Oct 17, 2011, 12:39:30 PM (11 years ago)

Author:

sam

Comment:

add a solver for Mike Acton's attempts

research/trig

@@ -30 +30 @@
 {{{
 #!latex
-\[\max_{x \in [0, \pi^2/4]}{\big\lvert\sin(\sqrt{y}) - \sqrt{y}Q(y)\big\rvert} = E\]
+\[\max_{y \in [0, \pi^2/4]}{\big\lvert\sin(\sqrt{y}) - \sqrt{y}Q(y)\big\rvert} = E\]
 }}}
 
@@ -37 +37 @@
 {{{
 #!latex
-\[\max_{x \in [0, \pi^2/4]}{\dfrac{\bigg\lvert\dfrac{\sin(\sqrt{y})}{\sqrt{y}} - Q(y)\bigg\rvert}{\dfrac{1}{|\sqrt{y}|}}} = E\]
+\[\max_{y \in [0, \pi^2/4]}{\dfrac{\bigg\lvert\dfrac{\sin(\sqrt{y})}{\sqrt{y}} - Q(y)\bigg\rvert}{\dfrac{1}{|\sqrt{y}|}}} = E\]
 }}}
 
@@ -44 +44 @@
 {{{
 #!latex
-\[\max_{x \in [0, \pi^2/4]}{\dfrac{\bigg\lvert\dfrac{\sin(\sqrt{y})}{\sqrt{y}} - 1 - yR(y)\bigg\rvert}{\dfrac{1}{|\sqrt{y}|}}} = E\]
+\[\max_{y \in [0, \pi^2/4]}{\dfrac{\bigg\lvert\dfrac{\sin(\sqrt{y})}{\sqrt{y}} - 1 - yR(y)\bigg\rvert}{\dfrac{1}{|\sqrt{y}|}}} = E\]
 }}}
 
@@ -51 +51 @@
 {{{
 #!latex
-\[\max_{x \in [0, \pi^2/4]}{\dfrac{\bigg\lvert\dfrac{\sin(\sqrt{y})-\sqrt{y}}{y\sqrt{y}} - R(y)\bigg\rvert}{\dfrac{1}{|y\sqrt{y}|}}} = E\]
+\[\max_{y \in [0, \pi^2/4]}{\dfrac{\bigg\lvert\dfrac{\sin(\sqrt{y})-\sqrt{y}}{y\sqrt{y}} - R(y)\bigg\rvert}{\dfrac{1}{|y\sqrt{y}|}}} = E\]
 }}}
 
@@ -70 +70 @@
 
 RemezSolver<6> solver;
-solver.Run(real::R_1 >> 400, real::R_PI_2 * real::R_PI_2, myfun, myerr, 15);
+solver.Run(real::R_1 >> 400, real::R_PI_2 * real::R_PI_2, myfun, myerr, 40);
 }}}
 
@@ -88 +88 @@
 {{{
 #!cpp
-a0 = -1.66666666666658080941942898789420724e-1;
-a1 = +8.33333333326271609442503773834687308e-3;
-a2 = -1.98412698200591143928364634696492885e-4;
-a3 = +2.75573160733868922065738227278330896e-6;
-a4 = -2.50518513021429359590028300127165228e-8;
-a5 = +1.60472959182597740337401201006549498e-10;
-a6 = -7.36458957326227991327065122848667046e-13;
+a0 = -1.666666666666580938362041558393413542600e-1;
+a1 = +8.333333333262715528278347301093116699226e-3;
+a2 = -1.984126982005911547055378498482154233331e-4;
+a3 = +2.755731607338689059680115311170244593349e-6;
+a4 = -2.505185130214293461336864464029272975945e-8;
+a5 = +1.604729591825977276746033955401272849354e-10;
+a6 = -7.364589573262279656101943192883347174447e-13;
+
+E = 1.098969630370672683831702893969063712485e-16;
 }}}
 
@@ -110 +112 @@
 {{{
 #!latex
-\[\max_{x \in [0, \pi^2/4]}{\dfrac{\bigg\lvert\dfrac{\sin(\sqrt{y})-\sqrt{y}}{y\sqrt{y}} - R(y)\bigg\rvert}{\bigg\lvert\dfrac{\sin(y)}{y\sqrt{y}}\bigg\rvert}} = E\]
+\[\max_{y \in [0, \pi^2/4]}{\dfrac{\bigg\lvert\dfrac{\sin(\sqrt{y})-\sqrt{y}}{y\sqrt{y}} - R(y)\bigg\rvert}{\bigg\lvert\dfrac{\sin(y)}{y\sqrt{y}}\bigg\rvert}} = E\]
 }}}
 
@@ -128 +130 @@
 
 RemezSolver<6> solver;
-solver.Run(real::R_1 >> 400, real::R_PI_2 * real::R_PI_2, myfun, myerr, 15);
+solver.Run(real::R_1 >> 400, real::R_PI_2 * real::R_PI_2, myfun, myerr, 40);
 }}}
 
 {{{
 #!cpp
-a0 = -1.666666666666666587374325845020415990185e-1;
-a1 = +8.333333333333133768001243698120735518527e-3;
-a2 = -1.984126984109960366729319073763957206143e-4;
-a3 = +2.755731915499171528179303925040423384803e-6;
-a4 = -2.505209340355388148617179634180834358690e-8;
-a5 = +1.605725287696319345779134635418774782711e-10;
-a6 = -7.535968124281960435283756562793611388136e-13;
+a0 = -1.666666666666618655330686951220600109093e-1;
+a1 = +8.333333333285509269011390197274270193963e-3;
+a2 = -1.984126982504390943378441670654599796999e-4;
+a3 = +2.755731659890484005079622148869365590255e-6;
+a4 = -2.505188017067512158000464673183918592305e-8;
+a5 = +1.604809231079007402834515514014751626424e-10;
+a6 = -7.373308642081174610234470417310065337523e-13;
+
+E = 1.536616934979294924070319645582179175464e-16;
 }}}
+
+=== Another example ===
+
+Trying to reproduce [http://www.insomniacgames.com/remez-exchange-algorithm/ Mike Acton]'s experimental result:
+
+{{{
+#!latex
+\[\max_{x \in [-1, 1]}{\dfrac{\big\vert\sin(\frac\pi{2}x) - P(x)\big\vert}{|\sin(\frac\pi{2}x)|}} = E\]
+}}}
+
+Using `P(x) = xQ(x²)`:
+
+{{{
+#!latex
+\[\max_{y \in [0, 1]}{\dfrac{\bigg\lvert\dfrac{\sin(\frac\pi{2}\sqrt{y})}{\sqrt{y}} - Q(y)\bigg\rvert}{\bigg\lvert\dfrac{\sin(\frac\pi{2}y)}{\sqrt{y}}\bigg\rvert}} = E\]
+}}}
+
+{{{
+#!cpp
+static real myfun(real const &y)
+{
+    real x = sqrt(y);
+    return sin(real::R_PI_2 * x) / x;
+}
+
+RemezSolver<4> solver;
+solver.Run(real::R_1 >> 400, real::R_1, myfun, myfun, 40);
+}}}
+
+{{{
+#!cpp
+a0 = +1.570796318452974170937444182514099057668;
+a1 = -6.459637106518004316895285560913086847441e-1;
+a2 = +7.968967893119537554369879233657335437244e-2;
+a3 = -4.673766402124326801818077428151491285886e-3;
+a4 = +1.514849803879821510688050931907848092814e-4;
+
+E = 5.310632770140865101487747414605249755889e-9;
+}}}