* Bug 16290 fixed [special_functions]: ellipj() introduced 92/21192/9
Samuel Gougeon [Sun, 29 Dec 2019 02:44:54 +0000 (03:44 +0100)]
  http://bugzilla.scilab.org/16290

  PDF ellipj() page: http://bugzilla.scilab.org/attachment.cgi?id=5039

Change-Id: I94c56e9730ff9bd517ab63e00fd4f09fa0f5981b

23 files changed:
scilab/CHANGES.md
scilab/modules/helptools/data/configuration/scilab_macros.txt
scilab/modules/helptools/etc/images_md5.txt
scilab/modules/helptools/images/_LaTeX_ellipj.xml_1.png [new file with mode: 0644]
scilab/modules/helptools/images/_LaTeX_histc.xml_1.png
scilab/modules/helptools/images/ellipj_1.png [new file with mode: 0644]
scilab/modules/helptools/images/ellipj_2.png [new file with mode: 0644]
scilab/modules/special_functions/help/en_US/amell.xml
scilab/modules/special_functions/help/en_US/delip.xml
scilab/modules/special_functions/help/en_US/ellipj.xml [new file with mode: 0644]
scilab/modules/special_functions/help/en_US/percentk.xml
scilab/modules/special_functions/help/fr_FR/amell.xml
scilab/modules/special_functions/help/fr_FR/delip.xml
scilab/modules/special_functions/help/ja_JP/amell.xml
scilab/modules/special_functions/help/ja_JP/delip.xml
scilab/modules/special_functions/help/ja_JP/percentk.xml
scilab/modules/special_functions/help/pt_BR/amell.xml
scilab/modules/special_functions/help/pt_BR/delip.xml
scilab/modules/special_functions/help/ru_RU/amell.xml
scilab/modules/special_functions/help/ru_RU/delip.xml
scilab/modules/special_functions/help/ru_RU/percentk.xml
scilab/modules/special_functions/macros/ellipj.sci [new file with mode: 0644]
scilab/modules/special_functions/tests/unit_tests/ellipj.tst [new file with mode: 0644]

index 13a2628..4e153b3 100644 (file)
@@ -124,6 +124,7 @@ Feature changes and additions
   - return the number of occurences of distinct entities found in the input array.
   - return distinct entities in their initial order (rather than sorted), with the `"keepOrder"` option.
   - consider all `Nan` values as the same one, with the `"uniqueNan"` option.
+* `ellipj()` function introduced, to compute the `sn`, `cn`, `dn`, `ns`, `nc` and `nd` Jacobi elliptic functions.
 
 
 Help pages:
@@ -215,12 +216,12 @@ Bug Fixes
 * [#15425](http://bugzilla.scilab.org/show_bug.cgi?id=15425): The Kronecker product `a.*.b` failed when `a` or `b` or both are hypermatrices, with one or both being polynomials or rationals.
 * [#15451](http://bugzilla.scilab.org/show_bug.cgi?id=15451): The code was not adapted to use `lucene 4.10` in Debian.
 * [#15514](http://bugzilla.scilab.org/show_bug.cgi?id=15514): The `set()` documentation page needed to be overhauled.
-* [#15522](http://bugzilla.scilab.org/show_bug.cgi?id=15522): `unique()` was not able to consider all Nan values as the same value. A `uniqueNan` option now allows it. 
+* [#15522](http://bugzilla.scilab.org/show_bug.cgi?id=15522): `unique()` was not able to consider all Nan values as the same value. A `uniqueNan` option now allows it.
 * [#15523](http://bugzilla.scilab.org/show_bug.cgi?id=15523): `%ODEOPTIONS(1)=2` didn't work with solvers 'rk' and 'rkf'
 * [#15248](http://bugzilla.scilab.org/show_bug.cgi?id=15248): `lsq()`was leaking memory.
 * [#15577](http://bugzilla.scilab.org/show_bug.cgi?id=15577): `edit` did not accept a line number as text, as with `edit linspace 21`.
 * [#15580](http://bugzilla.scilab.org/show_bug.cgi?id=15580): `det(sparse([],[]))` yielded an error.
-* [#15595](http://bugzilla.scilab.org/show_bug.cgi?id=15595): `unique()` was not able to return distinct values in their original order, without sorting them. A `keepOrder` option now allows it. 
+* [#15595](http://bugzilla.scilab.org/show_bug.cgi?id=15595): `unique()` was not able to return distinct values in their original order, without sorting them. A `keepOrder` option now allows it.
 * [#15668](http://bugzilla.scilab.org/show_bug.cgi?id=15668): `save(filename)` saved all predefined Scilab constants %e %pi etc.. (regression)
 * [#15715](http://bugzilla.scilab.org/show_bug.cgi?id=15715): `%nan` indices crashed Scilab.
 * [#15742](http://bugzilla.scilab.org/show_bug.cgi?id=15742): The `compatibility_functions` module should be merged in the `m2sci` one.
@@ -288,6 +289,8 @@ Bug Fixes
 * [#16272](http://bugzilla.scilab.org/show_bug.cgi?id=16272): `spzeros(0,n)` and `spzeros(n,0)` were different from `sparse(0,0)`.
 * [#16273](http://bugzilla.scilab.org/show_bug.cgi?id=16273): `calendar()` had no formated display mode.
 * [#16275](http://bugzilla.scilab.org/show_bug.cgi?id=16275): `fsolve(x0, fun, tol)` no longer took `tol` into account.
+* [#16290](http://bugzilla.scilab.org/show_bug.cgi?id=16290): The `cn`, `dn`, `ns`, `nc` and `nd` Jacobi elliptic functions were not available.
 * [#16293](http://bugzilla.scilab.org/show_bug.cgi?id=16293): Some demos run in step-by-step console mode(4) did not focus user's attention to the console to proceed.
 * [#16299](http://bugzilla.scilab.org/show_bug.cgi?id=16299): After `graypolarplot()`, `colorbar()` displayed an empty ungraduated color bar.
 * [#16303](http://bugzilla.scilab.org/show_bug.cgi?id=16303): log10(x) had wrong dimensions when x is an hypermatrix.
+*
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diff --git a/scilab/modules/helptools/images/_LaTeX_ellipj.xml_1.png b/scilab/modules/helptools/images/_LaTeX_ellipj.xml_1.png
new file mode 100644 (file)
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index fabd3db..24d35b4 100644 (file)
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diff --git a/scilab/modules/helptools/images/ellipj_1.png b/scilab/modules/helptools/images/ellipj_1.png
new file mode 100644 (file)
index 0000000..2e8ce47
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diff --git a/scilab/modules/helptools/images/ellipj_2.png b/scilab/modules/helptools/images/ellipj_2.png
new file mode 100644 (file)
index 0000000..ac076b2
Binary files /dev/null and b/scilab/modules/helptools/images/ellipj_2.png differ
index c410876..1d28e44 100644 (file)
@@ -61,6 +61,9 @@
                 <link linkend="delip">delip</link>
             </member>
             <member>
+                <link linkend="ellipj">ellipj</link>
+            </member>
+            <member>
                 <link linkend="percentsn">%sn</link>
             </member>
         </simplelist>
index 7c66b9b..3a13be4 100644 (file)
@@ -143,6 +143,9 @@ intg(0,1,f)    //OK since real solution!
                 <link linkend="amell">amell</link>
             </member>
             <member>
+                <link linkend="ellipj">ellipj</link>
+            </member>
+            <member>
                 <link linkend="percentsn">%sn</link>
             </member>
         </simplelist>
diff --git a/scilab/modules/special_functions/help/en_US/ellipj.xml b/scilab/modules/special_functions/help/en_US/ellipj.xml
new file mode 100644 (file)
index 0000000..c0f811c
--- /dev/null
@@ -0,0 +1,296 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<!--
+ * Copyright (C) 2019 - Samuel GOUGEON - Le Mans Université
+ *
+ * This file is hereby licensed under the terms of the GNU GPL v2.0,
+ * pursuant to article 5.3.4 of the CeCILL v.2.1.
+ * This file was originally licensed under the terms of the CeCILL v2.1,
+ * and continues to be available under such terms.
+ * For more information, see the COPYING file which you should have received
+ * along with this program.
+ *
+ -->
+<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink"
+          xmlns:svg="http://www.w3.org/2000/svg" xmlns:mml="http://www.w3.org/1998/Math/MathML"
+          xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org"
+          xml:lang="en" xml:id="ellipj">
+    <refnamediv>
+        <refname>ellipj</refname>
+        <refpurpose>Jacobi elliptic functions</refpurpose>
+    </refnamediv>
+    <refsynopsisdiv>
+        <title>Syntax</title>
+        <synopsis>
+        sn = ellipj(x, m)
+        [sn, cn] = ellipj(x, m)
+        [sn, cn, dn] = ellipj(x, m)
+        struc = ellipj(funames, x, m)
+        </synopsis>
+    </refsynopsisdiv>
+    <refsection>
+        <title>Arguments</title>
+        <variablelist>
+            <varlistentry>
+                <term>x</term>
+                <listitem>
+                    Array of real or complex numbers inside the fundamental rectangle
+                    defined by the elliptic integral.
+                    <para/>
+                </listitem>
+            </varlistentry>
+            <varlistentry>
+                <term>m</term>
+                <listitem>
+                    Scalar decimal number in <literal>[0,1]</literal>: Parameter of the
+                    elliptic integral.
+                    <para/>
+                </listitem>
+            </varlistentry>
+            <varlistentry>
+                <term>sn, cn, dn</term>
+                <listitem>
+                    Arrays of real or complex numbers, of the size of <varname>x</varname>:
+                    the x-element-wise values of the sn, cn and dn Jacobi elliptic functions.
+                    <para/>
+                </listitem>
+            </varlistentry>
+            <varlistentry>
+                <term>funames</term>
+                <listitem>
+                    vector of case-sensitive keywords among
+                    <literal>"cn", "dn", "sn", "nc", "nd", "ns"</literal>: Name(s) of Jacobi
+                    elliptic functions whose values must be computed and returned.
+                    <para/>
+                </listitem>
+            </varlistentry>
+            <varlistentry>
+                <term>struc</term>
+                <listitem>
+                    Structure with fields named with the elements of <varname>funames</varname>.
+                    <para/>
+                </listitem>
+            </varlistentry>
+        </variablelist>
+    </refsection>
+    <refsection>
+        <title>Description</title>
+        <para>
+            If <varname>x</varname> is real, <literal>sn = ellipj(x, m)</literal>
+            computes <varname>sn</varname> such that
+            <latex style="inline" alt="x = integral_0^{sn} du/sqrt{(1-u^2)(1-m.u^2)}">
+                \int_0^{sn}\frac{du}{\sqrt{(1-u^2)(1-m\,u^2)}}=x
+            </latex>
+        </para>
+        <para>
+            Then, if <literal>0 ≤ x ≤ x<subscript>M</subscript></literal> where
+            <literal>x<subscript>M</subscript></literal> is the position of the first
+            <literal>sn</literal> maximum,
+            we have <literal>x==delip(sn, sqrt(m))</literal>;
+            if <literal>-x<subscript>M</subscript> ≤ x ≤ 0</literal>,
+            we get <literal>x==-delip(-sn, sqrt(m))</literal>.
+        </para>
+        <para>
+            <literal>[sn,cn] = ellipj(x,m)</literal> and <literal>[sn,cn,dn] = ellipj(x,m)</literal>
+            additionally compute <literal>cn = cos(am)</literal> and
+            <literal>dn=(1-m.sn<superscript>2</superscript>)<superscript>1/2</superscript></literal>,
+            where <literal>am</literal> is the Jacobi elliptic amplitude function as computed with
+            <literal>amell()</literal>.
+        </para>
+        <para>
+            <literal>st = ellipj("dn", x, m)</literal> computes values of the <literal>dn</literal>
+            Jacobi elliptic function for given <varname>x</varname> values, and returns them in the
+            <literal>st.dn</literal> field.
+        </para>
+        <para>
+            Three other <literal>nc, nd, ns</literal> Jacobi
+            elliptic functions can be computed and returned with this syntax, as
+            <literal>nc = 1/cn</literal>, <literal>nd = 1/dn</literal>, and <literal>ns = 1/sn</literal>.
+        </para>
+        <para>
+            To compute values of all available Jacobi elliptic functions, we may use
+            <literal>st = ellipj(["cn" "nc" "dn" "nd" "sn" "ns"], x, m)</literal>.
+        </para>
+        <para>
+          <emphasis>Remark : </emphasis> for complex input numbers <varname>x</varname>,
+          <literal>amell()</literal> is used to compute the elliptic amplitude separately of
+          the real and imaginary parts of <varname>x</varname>. Then, addition formulae are used
+          to compute the complex <literal>sn</literal>, <literal>cn</literal> etc
+          [<ulink url="https://researchspace.auckland.ac.nz/bitstream/handle/2292/5042/390.pdf">Tee, 1997</ulink>].
+        </para>
+        <warning>
+            <para>
+                The relative numerical accuracy on <emphasis>sn</emphasis> roughly goes as
+                <literal>x * [1 to 50] * %eps</literal>.
+                Therefore, numerical results for <literal>x > 10<superscript>14</superscript></literal>
+                are not relevant.
+            </para>
+            <para>
+                <literal>ellipj(%inf, m)</literal> is set to NaN.
+            </para>
+        </warning>
+    </refsection>
+    <refsection>
+        <title>Examples</title>
+        <para>
+            With some real input numbers:
+        </para>
+        <programlisting role="example"><![CDATA[
+sn = ellipj(0:0.5:2, 0.4)
+delip(sn, sqrt(0.4))
+
+[sn, cn, dn] = ellipj(-1:0.5:1.5, 0.4)
+ ]]></programlisting>
+        <screen><![CDATA[
+--> sn = ellipj(0:0.5:2, 0.4)
+ sn  =
+   0.   0.4724822   0.8109471   0.9767495   0.9850902
+
+--> delip(sn, sqrt(0.4))
+ ans  =
+   0.   0.5   1.   1.5   1.5550387
+
+--> [sn, cn, dn] = ellipj(-1:0.5:1.5, 0.4)
+ sn  =
+  -0.8109471  -0.4724822   0.   0.4724822   0.8109471   0.9767495
+
+ cn  =
+   0.5851194   0.8813402   1.   0.8813402   0.5851194   0.2143837
+
+ dn  =
+   0.8584555   0.9543082   1.   0.9543082   0.8584555   0.786374
+]]></screen>
+        <para/>
+        <programlisting role="example"><![CDATA[
+// Input data
+x = linspace(-6,6,300)';
+m = [0.2 0.4 0.6 0.8 0.9];
+f = ["sn" "dn" "cn" "nd"];
+
+// Computation
+st = struct();
+for i = 1:length(m)
+    st(i) = ellipj(f,x,m(i));
+end
+
+// Plot
+clf
+for i = 1:size(f,"*")
+    subplot(2,2,i)
+    y = matrix(list2vec(st(f(i))), -1, length(m));
+    plot2d(x, y,[1 14 28 18 6])
+    title(f(i), "fontsize",4)
+    xgrid(color("grey75"),1,7)
+end
+
+legend(gca(), msprintf("m =%4.1f\n",m'),1, %f);
+lg = gce();
+lg.legend_location = "by_coordinates";
+lg.position = [-0.095 -0.08];
+ ]]></programlisting>
+        <scilab:image><![CDATA[
+            // Input data
+            x = linspace(-6,6,300)';
+            m = [0.2 0.4 0.6 0.8 0.9];
+            f = ["sn" "dn" "cn" "nd"];
+
+            // Computation
+            st = struct();
+            for i = 1:length(m)
+                st(i) = ellipj(f,x,m(i));
+            end
+
+            // Plot
+            clf
+            for i = 1:size(f,"*")
+                subplot(2,2,i)
+                y = matrix(list2vec(st(f(i))), -1, length(m));
+                plot2d(x, y,[1 14 28 18 6])
+                title(f(i), "fontsize",4)
+                xgrid(color("grey75"),1,7)
+            end
+
+            legend(gca(), msprintf("m =%4.1f\n",m'),1, %f);
+            lg = gce();
+            lg.legend_location = "by_coordinates";
+            lg.position = [-0.095 -0.08];
+
+            gcf().axes_size = [850 500];
+            ]]>
+        </scilab:image>
+        <para>
+            With some complex input numbers:
+        </para>
+        <programlisting role="example"><![CDATA[
+x = linspace(-6,6,300)';
+m = 0.1:0.1:0.9;
+sn = [];
+for i = 1:length(m)
+    sn(:,i) = ellipj(x+1.4*%i, m(i));
+end
+
+clf
+plot2d(imag(sn), real(sn))
+title("sn(-6:6 + 1.4i)", "fontsize",4)
+xlabel("imag(sn)", "fontsize",2);
+ylabel("real(sn)", "fontsize",2);
+colordef(gcf(),"none")
+gca().data_bounds(1:2) = [-5 5];
+gca().tight_limits(1) = "on";
+isoview
+legend(gca(), msprintf("m =%4.1f\n",m'),"out_upper_right");
+     ]]></programlisting>
+        <para/>
+        <scilab:image><![CDATA[
+            x = linspace(-6,6,300)';
+            m = 0.1:0.1:0.9;
+            sn = [];
+            for i = 1:length(m)
+                sn(:,i) = ellipj(x+1.4*%i, m(i));
+            end
+
+            clf
+            plot2d(imag(sn), real(sn))
+            title("sn(-6:6 + 1.4i)", "fontsize",4)
+            xlabel("imag(sn)", "fontsize",2);
+            ylabel("real(sn)", "fontsize",2);
+            colordef(gcf(),"none")
+            gca().data_bounds(1:2) = [-5 5];
+            gca().tight_limits(1) = "on";
+            isoview
+            legend(gca(), msprintf("m =%4.1f\n",m'),"out_upper_right");
+            gcf().axes_size = [600 350];
+    ]]> </scilab:image>
+    </refsection>
+    <refsection role="see also">
+        <title>See also</title>
+        <simplelist type="inline">
+            <member>
+                <link linkend="amell">amell</link>
+            </member>
+            <member>
+                <link linkend="delip">delip</link>
+            </member>
+            <member>
+                <link linkend="percentk">%k</link>
+            </member>
+            <member>
+                <ulink url="https://researchspace.auckland.ac.nz/bitstream/handle/2292/5042/390.pdf">
+                Tee, Garry J.</ulink>: Continuous branches of inverses of the 12 Jacobi elliptic
+                functions for real argument”. 1997
+            </member>
+
+
+        </simplelist>
+    </refsection>
+    <refsection role="history">
+        <title>History</title>
+        <revhistory>
+            <revision>
+                <revnumber>6.1.0</revnumber>
+                <revdescription>
+                    ellipj() introduced.
+                </revdescription>
+            </revision>
+        </revhistory>
+    </refsection>
+</refentry>
index eb8ba3f..f348146 100644 (file)
@@ -70,6 +70,9 @@ m = 0.4;
             <member>
                 <link linkend="delip">delip</link>
             </member>
+            <member>
+                <link linkend="ellipj">ellipj</link>
+            </member>
         </simplelist>
     </refsection>
 </refentry>
index 8689884..5d8fdc9 100644 (file)
@@ -50,6 +50,9 @@
                 <link linkend="delip">delip</link>
             </member>
             <member>
+                <link linkend="ellipj">ellipj</link>
+            </member>
+            <member>
                 <link linkend="percentsn">%sn</link>
             </member>
         </simplelist>
index 244d63a..f1fadc4 100644 (file)
@@ -128,6 +128,9 @@ intg(0,1,f)    // OK puisque la solution est réelle !
                 <link linkend="amell">amell</link>
             </member>
             <member>
+                <link linkend="ellipj">ellipj</link>
+            </member>
+            <member>
                 <link linkend="percentsn">%sn</link>
             </member>
         </simplelist>
index cf1accc..43ff203 100644 (file)
@@ -62,6 +62,9 @@
                 <link linkend="delip">delip</link>
             </member>
             <member>
+                <link linkend="ellipj">ellipj</link>
+            </member>
+            <member>
                 <link linkend="percentsn">%sn</link>
             </member>
         </simplelist>
index a256eb0..6984b1c 100644 (file)
@@ -144,7 +144,7 @@ intg(0,1,f)    //OK since real solution!
                 <link linkend="amell">amell</link>
             </member>
             <member>
-                <link linkend="delip">delip</link>
+                <link linkend="ellipj">ellipj</link>
             </member>
             <member>
                 <link linkend="percentsn">%sn</link>
index 4fe457a..9c81e8c 100644 (file)
@@ -69,6 +69,9 @@ m = 0.4;
             <member>
                 <link linkend="delip">delip</link>
             </member>
+            <member>
+                <link linkend="ellipj">ellipj</link>
+            </member>
         </simplelist>
     </refsection>
 </refentry>
index 633e55d..5176502 100644 (file)
@@ -62,6 +62,9 @@
                 <link linkend="delip">delip</link>
             </member>
             <member>
+                <link linkend="ellipj">ellipj</link>
+            </member>
+            <member>
                 <link linkend="percentsn">%sn</link>
             </member>
         </simplelist>
index b447190..085b9ce 100644 (file)
@@ -140,6 +140,9 @@ intg(0,1,f)    //OK, desde que a solução seja real!
                 <link linkend="amell">amell</link>
             </member>
             <member>
+                <link linkend="ellipj">ellipj</link>
+            </member>
+            <member>
                 <link linkend="percentsn">%sn</link>
             </member>
         </simplelist>
index ce0db6b..cf107db 100644 (file)
@@ -61,6 +61,9 @@
                 <link linkend="delip">delip</link>
             </member>
             <member>
+                <link linkend="ellipj">ellipj</link>
+            </member>
+            <member>
                 <link linkend="percentsn">%sn</link>
             </member>
         </simplelist>
index ee0533e..4163019 100644 (file)
@@ -143,6 +143,9 @@ intg(0,1,f)    //OK так как решение вещественное!
                 <link linkend="amell">amell</link>
             </member>
             <member>
+                <link linkend="ellipj">ellipj</link>
+            </member>
+            <member>
                 <link linkend="percentsn">%sn</link>
             </member>
         </simplelist>
index b004e14..a455698 100644 (file)
@@ -73,6 +73,9 @@ m = 0.4;
             <member>
                 <link linkend="delip">delip</link>
             </member>
+            <member>
+                <link linkend="ellipj">ellipj</link>
+            </member>
         </simplelist>
     </refsection>
 </refentry>
diff --git a/scilab/modules/special_functions/macros/ellipj.sci b/scilab/modules/special_functions/macros/ellipj.sci
new file mode 100644 (file)
index 0000000..60c6f50
--- /dev/null
@@ -0,0 +1,157 @@
+// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
+//
+// Copyright (C) 2019 - Samuel GOUGEON - Le Mans Université
+//
+// This file is hereby licensed under the terms of the GNU GPL v2.0,
+// pursuant to article 5.3.4 of the CeCILL v.2.1.
+// This file was originally licensed under the terms of the CeCILL v2.1,
+// and continues to be available under such terms.
+// For more information, see the COPYING file which you should have received
+// along with this program.
+
+function varargout = ellipj(varargin)
+    //           sn = ellipj(x, m)  // Replaces %sn(x,m)
+    //     [sn, cn] = ellipj(x, m)  // Returns the cn function in addition to sn
+    // [sn, cn, dn] = ellipj(x, m)  // Returns dn function in addition to sn and cn
+    //           st = ellipj(funNames, x, m)
+    //               // Returns the Jacobi functions named in the funNames
+    //               //  vector of strings among "sn" "cn" "dn" "ns" "nc" "nd".
+    //               //  Values are returned through the st structure whose fields
+    //               //  are the elements of funNames.
+    //               //  Later, this syntax could be easily extendable to any
+    //               //  of the other non trivial "cd" "cs" "dc" "ds" "sc"
+    //               //  and "sd" Jacobi functions.
+
+    fname = "ellipj"
+    lhs = argn(1)
+    funs = []   // names of Jacobi functions to be returned
+    shift = 0   // input argument shift, for error messages about x or m
+
+    // CHECKING ARGUMENTS
+    // ==================
+    in = varargin
+    sizin = length(varargin)
+    if sizin < 2 | sizin > 3 then
+        msg = _("%s: Wrong number of input arguments: %d or %d expected.\n")
+        error(msprintf(msg, fname, 2, 3))
+    end
+    if sizin==3 then
+        [funs, k]= unique(in(1))
+        funs = in(1)(gsort(k,"g","i"))
+        in(1) = null()
+        if lhs > 1
+            msg = _("%s: Wrong number of output arguments: %d expected.\n")
+            error(msprintf(msg, fname, 1))
+        end
+        shift = 1
+        if type(funs) <> 10
+            msg = _("%s: Argument #%d: Text expected.\n")
+            error(msprintf(msg, fname, 1))
+        end
+        if length(grep(["cn" "dn" "sn" "nc" "nd" "ns"], funs)) <> size(funs,"*")
+            msg = _("%s: Argument #%d: Must be in the set {%s}.\n")
+            error(msprintf(msg, fname, 1, "''cn'' ''dn'' ''sn'' ''nc'' ''nd'' ''ns''"))
+        end
+    else
+        if lhs > 3
+            msg = _("%s: Wrong number of output arguments: %d to %d expected.\n")
+            error(msprintf(msg, fname, 1, 3))
+        end
+    end
+    // x
+    x = in(1)
+    if type(x) <> 1 then
+        msg = _("%s: Argument #%d: Decimal or complex number expected.\n")
+        error(msprintf(msg, fname, 1+shift))
+    end
+    sx = size(x)
+    x = x(:)
+    // m
+    m = in(2)
+    if type(m) <> 1 | ~isreal(m,0) then
+        msg = _("%s: Argument #%d: Decimal number expected.\n")
+        error(msprintf(msg, fname, 2+shift))
+    end
+    if or(m < 0) | or(m > 1) then
+        msg = _("%s: Argument #%d: Must be in the interval [%d, %d].\n")
+        error(msprintf(msg, fname, 2+shift, 0, 1))
+    end
+
+    // PROCESSING
+    // ==========
+    [n1,n2] = size(x);
+    n = n1*n2;
+    a = amell(real(x), sqrt(m));
+    s = sin(a);
+    c = cos(a);
+    d = sqrt(ones(n1,n2) - m*s.*s);
+    xReal = isreal(x,0)
+    if ~xReal then
+        m1 = 1 - m;
+        a1 = amell(imag(x), sqrt(m1));
+        s1 = sin(a1);
+        c1 = cos(a1);
+        d1 = sqrt(ones(n1,n2) - m1*s1.*s1);
+        N = (c1.*c1 + m*s.*s.*s1.*s1);
+    end
+    kinf = isinf(x)
+
+    // sn
+    // --
+    if funs==[] | grep(funs,["sn" "ns" "dn" "nd"]) <> [] then
+        if xReal
+            sn = s
+        else
+            sn = (s.*d1 + %i*c.*d.*s1.*c1) ./ N;
+        end
+        if ~isreal(sn) & isreal(sn,0) then
+            sn = real(sn)
+        end
+        sn(kinf) = %nan
+        sn = matrix(sn, sx)
+    end
+    // cn
+    // --
+    if lhs > 1 | grep(funs,["cn" "nc"]) <> [] then
+        if xReal
+            cn = c
+        else
+            cn = (c.*c1 - %i*s.*d.*s1.*d1) ./ N;
+        end
+        if ~isreal(cn) & isreal(cn,0) then
+            cn = real(cn)
+        end
+        cn(kinf) = %nan
+        cn = matrix(cn, sx)
+    end
+    // dn
+    // --
+    if lhs > 2 | grep(funs,["dn" "nd"]) <> [] then
+        dn = sqrt(1 - m*sn.*sn);
+        if ~isreal(dn) & isreal(dn,0) then
+            dn = real(dn)
+        end
+    end
+    // nc, nd, ns
+    // ----------
+    if funs <> [] then
+        if grep(funs, "nc") <> [], nc = 1 ./ cn, end
+        if grep(funs, "nd") <> [], nd = 1 ./ dn, end
+        if grep(funs, "ns") <> [], ns = 1 ./ sn, end
+    end
+
+    // OUTPUT
+    // ======
+    if funs <> [] then
+        st = struct()
+        for f = funs(:)'
+            execstr("st(f) = "+f)
+        end
+        varargout = list(st)
+    else
+        varargout = list()
+        varargout(1) = sn
+        if lhs > 1, varargout(2) = cn, end
+        if lhs > 2, varargout(3) = dn, end
+    end
+endfunction
diff --git a/scilab/modules/special_functions/tests/unit_tests/ellipj.tst b/scilab/modules/special_functions/tests/unit_tests/ellipj.tst
new file mode 100644 (file)
index 0000000..a9cfe4a
--- /dev/null
@@ -0,0 +1,171 @@
+// =============================================================================
+// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
+//
+// Copyright (C) 2019 - Samuel GOUGEON - Le Mans Université
+//
+//  This file is distributed under the same license as the Scilab package.
+// =============================================================================
+// <-- CLI SHELL MODE -->
+// <-- NO CHECK REF -->
+// <-- ENGLISH IMPOSED -->
+
+// --------------------------------
+// unit tests of ellipj() functions
+// --------------------------------
+
+// Check the result's size
+// =======================
+for o = list([], -3, rand(1,10), rand(10,1), rand(10,12), rand(3,4,2))
+    assert_checkequal(size(ellipj(o,0.1)), size(o));
+end
+
+// Special input value
+// ===================
+[sn,cn,dn] = ellipj(%inf, 0.5)
+assert_checkequal([sn cn dn], [%nan %nan %nan]);
+
+// ========
+// sn(real)
+// ========
+ref = [// Reference values from  W o l f r a m   A l p h a
+// x         m = 0.7         rtol
+   0.   0                   0
+   1.   0.7867623461751605  5e-16
+   2.   0.9991470517177337  5e-16
+   3.   0.8496055646182321  5e-16
+   4.   0.1497633260257520  1e-15
+   5.  -0.7090590533602067  5e-16
+   6.  -0.9922509616227788  1e-15
+   7.  -0.8991504331040869  %eps
+   8.  -0.2939180408456228  %eps
+   9.   0.6153795020583048  %eps
+   10.  0.9780732803575124  %eps
+   ];
+for i = 1:size(ref,1)
+    assert_checkalmostequal(ellipj(ref(i), 0.7), ref(i,2), ref(i,3));
+    assert_checkalmostequal(ellipj(-ref(i), 0.7), -ref(i,2), ref(i,3));
+end
+// Powers of 10:
+// x          m = 0.05         rtol          m = 0.5      rtol        m = 0.95        rtol
+ref = [ // Reference values from  W o l f r a m   A l p h a
+   1      0.8377707054731771  2e-16   0.8030018248956439, 5e-16   0.7658522624824511  5e-16
+   10    -0.4378729750655788  2e-15   0.8588125059527787, 5e-16  -0.934978691527766   5e-16
+   100   -0.9742526048177048  1e-15   0.1196019281577434, 2e-13  -0.811531427200948   1e-14
+   1e3    0.7482430909578792  5e-14  -0.8878321984811046, 1e-13  -0.437315386732894   2e-12
+   1e4    0.8624182984404760  2e-13   0.7384500010693715, 2e-12  -0.818571113592376   5e-12
+   1e5    0.7988835860291854  2e-12  -0.9402218551681980, 5e-12  -0.2618682783394857  5e-10
+   1e6   -0.2329089936053463  2e-10   0.8547923424610843, 1e-10  -0.998669351872119   2e-11
+   1e7    0.7352980451000191  5e-10  -0.0220307669223278, 1e-7   -0.989185882675048   1e-9
+   1e8   -0.9430613258095234  1e-9    0.2176954461361798, 1e-7   -0.3661544064410038  2e-7
+   1e9   -0.2890611571169084  1e-7    0.9690463387009056, 5e-8   -0.972228411178568   2e-7
+   1e10  -0.2448382176246635  2e-6   -0.2129864775603840, 1e-5   -0.992002095107755   1e-6
+   1e11   0.6477036508616879  5e-6   -0.9771767833683986, 5e-6    0.3888373760830963  5e-4
+   1e12  -0.6485071868288868  5e-5    0.6266882256650892, 2e-4    0.946251142823198   1e-4
+   1e13  -0.6565030164219840  5e-4   -0.3692666652213536, 5e-3   -0.2663201863292489  5e-2
+   1e14  -0.7321933463148297  5e-3    0.1188550465207018, 2e-1    0.998513062880542   2e-3
+   ];
+ref(:,[3 5 7]) = ref(:,[3 5 7])*2; // Linux is a bit less accurate than Windows
+for i = 1:size(ref,1)
+    assert_checkalmostequal(ellipj(ref(i), 0.05), ref(i,2), ref(i,3))
+    assert_checkalmostequal(ellipj(ref(i), 0.50), ref(i,4), ref(i,5))
+    assert_checkalmostequal(ellipj(ref(i), 0.95), ref(i,6), ref(i,7))
+    // oddness
+    assert_checkalmostequal(ellipj(-ref(i), 0.05), -ref(i,2), ref(i,3))
+    assert_checkalmostequal(ellipj(-ref(i), 0.50), -ref(i,4), ref(i,5))
+    assert_checkalmostequal(ellipj(-ref(i), 0.95), -ref(i,6), ref(i,7))
+end
+
+// ========
+// cn(real)
+// ========
+ref = [// Reference values from  W o l f r a m   A l p h a
+// x         m = 0.7         rtol
+   0.   1                   0
+   1.   0.61725603329652182 1e-15
+   2.   0.04129369254208586 5e-13
+   3.  -0.5274186046867658  2e-15
+   4.  -0.9887218750375175  2e-16
+   5.  -0.7051491039829290  5e-16
+   6.  -0.1242498658295887  5e-14
+   7.   0.4376396904403587  5e-16
+   8.   0.9558306258252405  %eps
+   9.   0.7882309740466133  %eps
+   10.  0.2082610339230434  2e-15
+   ];
+for i = 1:size(ref,1)
+    assert_checkalmostequal(ellipj("cn", ref(i), 0.7).cn,  ref(i,2), ref(i,3));
+    assert_checkalmostequal(ellipj("cn",-ref(i), 0.7).cn,  ref(i,2), ref(i,3));
+end
+
+// =====================
+// Complex input numbers
+// =====================
+// sn
+// --
+i = %i;
+ref = [// Reference values from  W o l f r a m   A l p h a
+// x+i       m = 0.2                             rtol
+   0.   1.240256801726044*i                      1e-15
+   1.   1.244235518007841+ 0.5361363537284757*i  %eps
+   2.   1.357582954109075-0.2637195472417189*i   2e-15
+   3.   0.5532930870698886-1.132321462346316*i   1e-15
+   4.  -1.012560212495065-0.8336903517823439*i   2e-15
+   5.  -1.393091630940202+0.01603358285570428*i  1e-12
+   6.  -0.9703342870049057+0.8729818250007476*i  2e-15
+   7.   0.6181470471828112+1.104123494298855*i   5e-15
+   8.   1.3661439825585795+0.229895115817073*i   2e-15
+   9.   1.221464554049673-0.5745426411640073*i   2e-15
+   10. -0.07691854949353121-1.238248750082493*i  5e-15
+   ];
+ref(:,3) = ref(:,3)*2;   // for Linux
+for k = 1:size(ref,1)
+    assert_checkalmostequal(ellipj(ref(k)+i,  0.2),  ref(k,2), ref(k,3));
+    assert_checkalmostequal(ellipj(-ref(k)-i, 0.2),  -ref(k,2), ref(k,3));
+    assert_checkalmostequal(ellipj(ref(k)-i,  0.2),  conj(ref(k,2)), ref(k,3));
+    assert_checkalmostequal(ellipj(-ref(k)+i,  0.2), -conj(ref(k,2)), ref(k,3));
+end
+
+// cn
+// --
+ref = [// Reference values from  W o l f r a m   A l p h a
+// x+i       m = 0.2                             rtol
+   0.   1.593184526107292                         5e-16
+   1.   0.7411843154734415-0.90001890201640132*i  1e-15
+   2.  -0.3745351729218371-0.9559079837754972 *i  2e-15
+   3.  -1.4689832461724766-0.42648930073878082*i  2e-15
+   4.  -1.1149172184787264+0.75715189053017862*i  1e-15
+   5.   0.02302611466014416+0.9700399055568715*i  1e-12
+   6.   1.16253366655224664+0.7286534756817488*i  2e-15
+   7.   1.43624591822981084-0.4752046074165396*i  5e-15
+   8.   0.3273265481638554-0.9595000187271159*i   1e-15
+   9.  -0.7908838324816837-0.8873407726262612*i   1e-15
+   10. -1.590888982052295+0.05986860104195037*i   5e-15
+   ];
+ref(:,3) = ref(:,3)*5; // for Linux
+for k = 1:size(ref,1)
+    assert_checkalmostequal(ellipj("cn",ref(k)+i,  0.2).cn,  ref(k,2), ref(k,3));
+    assert_checkalmostequal(ellipj("cn",-ref(k)-i, 0.2).cn,  ref(k,2), ref(k,3));
+    assert_checkalmostequal(ellipj("cn",ref(k)-i,  0.2).cn,  conj(ref(k,2)), ref(k,3));
+    assert_checkalmostequal(ellipj("cn",-ref(k)+i, 0.2).cn,  conj(ref(k,2)), ref(k,3));
+end
+
+
+// ====================
+// Check error messages
+// ====================
+msg = "ellipj: Wrong number of input arguments: 2 or 3 expected."
+assert_checkerror("ellipj()", msg);
+assert_checkerror("ellipj(1, 0.1, 2, 4)", msg);
+assert_checkerror("ellipj(""ns"", 0.1, 2, 4)", msg);
+msg = "ellipj: Wrong number of output arguments: 1 expected.";
+assert_checkerror("[a,b]=ellipj(""sn"",3,0.1)", msg);
+msg = "ellipj: Argument #1: Decimal or complex number expected.";
+assert_checkerror("ellipj(%t, 0.2)", msg);
+msg = "ellipj: Argument #2: Must be in the interval [0, 1].";
+assert_checkerror("ellipj(1, 1.2)", msg);
+msg = "ellipj: Argument #1: Text expected."
+assert_checkerror("ellipj(1, 0.1, 2)", msg);
+msg = "ellipj: Argument #2: Decimal or complex number expected.";
+assert_checkerror("ellipj(""sn"", %t, 0.2)", msg);
+msg = "ellipj: Argument #3: Must be in the interval [0, 1].";
+assert_checkerror("ellipj(""sn"",1, 1.2)", msg);