Summary: First IMaCh version using praxis method and adapted C code from Gegenfürtner

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+2024-04-24  Nicolas Brouard   <brouard@ined.fr>
+
+       * (Module): This version comes late after having tested
+       successfully the praxis C version of Buckhardt.  But Buckardt's
+       version was difficult to read and Gegenfurtner's version had a few
+       typos which made its results less reliable than Buckhardt's
+       results.  The most important work consisted in retyping the Brent
+       original PRAXIS program written in Algol W (published with errors,
+       ommitting the transposition of matrix V before its QR reduction
+       from Golub. I used the recent "awe" compiler from Gkynn Webster.
+       The awe library had errors, for example in arc tangent function
+       which have been fixed.
+
+       The main objective was to get identical results with the three
+       versions: (1) Algol W, (2) Buckhardt'C version as well (3)
+       Gegenfürtner C versions on the various test functions published by
+       Brent in 1973 in Algol W.
+
+       Also, in order to compare them, the random function had to produce
+       the same sequence for the 3 softwares. The random function used in
+       imach corresponds to original Brent's random function written in
+       Algol W.  Other point, in Algol W, the arrays of dimension n are
+       'normal' mathematical arrays starting from 1 to n. But this is a
+       real issue in C where, by default, arrays are starting from 0 to
+       n-1. In Buckhardt, as well as in Gegenfürtner C code, it can be
+       seen that authors while trying to mimick original Brent Algol W
+       code are hesitating by changing either a loop originally from 1 to
+       n in a loop from 0 to n-1, or keeping Brent's loop from 1 to n and
+       shifting the index from original X(I) in Algol W to x[i-1] in C.
+       But as IMaCh is using, since the beginning, the Numerical Recipes
+       functions vector or matrix, I changed Geggenfürtner code to mimick
+       the original Algol W arrays. Thus the X(I) is translated in C as
+       x[i] which minimizes the errors. The Golub QR algorithm was
+       published in Algol with overflow errors which were reproduced in
+       Brent's Algol W code. Buckhardt code fixed these errors which are
+       much more problematic in C than in Algol W.  Thus Buckhardt code
+       seems very safe, but i haven't chosen it for IMaCh because the C
+       style is horrible and almost unreadable compared to Gegenfürtner
+       CO code which is very close to Brent's original. Also what makes
+       Buckhardt code more difficult to read is, instead of passing the
+       minimum of parameters in the functions calls, as it is in Algol
+       Brent's code or Gegenfürtner's code, the list of parameters is
+       high. For example, the flin function LONG REAL PROCEDURE FLIN
+       (LONG REAL VALUE L) has only one parameter in Algol W, the
+       Gegenfürtner flin function had two parameters: static double
+       flin(l, j) double l; { int i; double tflin[N];} but Buckhardt
+       function has 14 parameters which makes the code unreadable and
+       useless. Gegenfürtner used a lot of static variables or functions
+       which I tried to minimize. Also in Gegefürtner, array dimensions
+       were fixed to N. In my adaptation the flin is static double
+       flin(double l, int j) and the parameter used are global variables.
+
+2023-06-14  Nicolas Brouard   <brouard@ined.fr>
+
+       * imach.c (Module): Testing if conjugate gradient could be quicker
+       when lot of variables POWELLORIGINCONJUGATE
+
 2023-05-23  Nicolas Brouard   <brouard@ined.fr>
 
        * imach.c (Module): Fixed PROB_r