shaw.ca>
Subject: Re: Possibility/ease of doing some vector ops in Maxima
Newsgroups: gmane.comp.mathematics.maxima.general
Date: 2002-09-19 01:00:07 GMT
Hi,
(...) At the Maxima prompt, type:
f:x*y*(x-y)^2;
v:[x,y];
g:makelist(diff(f,v[i]),i,1,2);
The first line is your function (...),
the second is a list of variables,
and the third line is the gradient (a list (vector) with elements being those of grad(f)).
I hope this helps, and gives you a few ideas...
Dave Holmgren
------------------
definition of gradient g of function f in pseudocode:
f(x,y):=x*y*(x-y)^2;
g:=diff(f,x)*i + diff(f,y)*j;
Complex number
Function: atan (x)
- Arc Tangent.
Function: atan2 (y, x)
- yields the value of atan(y/x) in the interval -%pi to %pi.
theta= carg(z)
The complex argument of z is an angle theta in (-%pi, %pi]
r exp (theta %i) = z
r = abs(z) is the complex magnitude of z
(%i1) c:0;
(%o1) 0
(%i2) atan(c);
(%o2) 0
(%i5) atan2(imagpart(c),realpart(c));
atan2(0,0) has been generated.
-- an error. To debug this try debugmode(true);
(%i6) carg(c);
(%o6) 0
Polar form :
(%i2) m(r,a):=r * exp(2*%i*%pi*a);
(%o2) c:m(1,0.5);
(%o2) -1
(%i62) carg(b)/(2*%pi);
(%o62) 1/2
(%i63) abs(b);
(%o63) 1
(%i9) z:zx+zy*%i;
(%o9) %i zy + zx
(%i10) expand(z^2);
(%o10) - zy^2 + 2 %i zx zy + zx^2
(%i11) c:cx+cy*%i;
(%o11) %i cy + cx
(%i12) expand(c+z^2);
(%o12) - zy^2 + 2 %i zx zy + zx^2 + %i cy + cx
(%i13) realpart(c+z^2);
(%o13) - zy^2 + zx^2 + cx
(%i14) imagpart(c+z^2);
(%o14) 2 zx zy + cy
(%i15) abs(z+c);
(%o15) sqrt((zy + cy)^2 + (zx + cx)^2 )
(%i16) carg(z);
(%o16) atan2(zy, zx)
(%i17) f(z):=z*z+c;
(%o17) f(z) := z z + c
(%i18) realpart(f(z));
(%o18) - zy2 + zx2 + cx
rectform(z*z+c);
(%i9) c:-.75 +.0001*%i;
(%o9) 1.0*10^-4*%i-0.75
(%i10) carg(c);
(%o10) 3.14145932025725
z:0+0*%i;
c:-2+0*%i;
for m:1 thru 20 step 1 do
block(
display(z, carg(z))
z:z*z+c
);
(%i1) solve([(c^2+c)^2+c=c^2+c], [c]);
(%o1) [c=-2,c=0]
Modulus of complex number :
plot3d(sqrt(u*u+v*v),[u,-5,5],[v,-7,7]);
the same in different form :
plot3d(cabs(u+v*%i),[u,-5,5],[v,-7,7]);
complex number in maxima byZdzislaw Meglicki
Prime factors:
a:ifactors(8);
[[2,3]]
(%i23) b:a[1][1];
(%o23) 2
(%i24) length(a);
(%o24) 1
save data (result of your computations) to file ( in the form of Lisp expressions) :
save("iim20.lisp",xy);
load it :
load("iim20.lisp");
Polynomials:
here P is a polynomial in one variable x
fP:factor(P);
part(fP,1); /* first factor */
rat(P, x); /* Factoring polynomials */
hipow(a,x); /* degree of expanded polynomial */
hipow(expand(P),x); /* degree of polynomial if non expanded */
(explanation by Stavros Macrakis )
Note that hipow is purely syntactic -- it looks for the highest *explicit*
power of x in the expression:
hipow((x^2-1)*(x-1),x) => 2 (not 3)
hipow((x^2-1)/(x-1),x) => 2 (not 1)
You may want to expand your expression with 'rat' first:
hipow(rat((x^2-1)*(x-1)),x) => 3
hipow(rat((x^2-1)/(x-1)),x) => 1
If your expression is in fact not a polynomial, you might be surprised by
the results, though they are correct by the syntactic definition:
hipow(1/x,x) => -1 (Maxima treats 1/x as x^(-1) )
hipow(1/(x^2-1),x); => 2 (not -2)
hipow(sin(x^2),x); => 2
/* gives a list of coefficients */
give_coefficients(P,x):=block
( P:expand(P), /* if not expanded */
degree:hipow(P,x),
a:makelist(coeff(P,x,degree-i),i,0,degree));
c_list:give_coefficients(P,x);
/* find max coeefficient */
c_max:lmax(c_list);
find its binary size :
log2(x) := log(x) / log(2);
log2(c_max);
If I'm not wrong fpprec should be greater then log2(c_max) ( for bfallroots)
Give_prec(P,x):=
ceiling(log2(lmax(give_coefficients(P,x))));
find all roots:
fpprec:32;
fpprintprec:8;
roots:bfallroots(%i*P);
(explanation be Leo Butler :)
(%i2) load(mnewton);
(%o2)
"/home/work/maxima/sandbox/maxima-current-release/share/contrib/mnewton.mac"
(%i3) g(x):=x^5-x+1;
(%o3) g(x):=x^5-x+1
(%i4) (fpprec:50, newtonepsilon:bfloat(10^(-fpprec+5)),
mnewton(g(x),x,-2));
(%o4) [[x = -1.1673039782614186842560458998548421807205603715255b0]]
Setting newtonepsilon to a bfloat means the computations are done
in bfloat. Use
? mnewton
to see the documentation.
"if I have a polynomial in the variable x, for instance
p:(a*x-b)*(3*x-2)+(c*x+d)*(1-2*x);
I can get the coefficients of the powers of x in the following way:
p_cre:rat(p,x);
c0:coeff(p_cre,x,0);
c1:coeff(p_cre,x,1);
c2:coeff(p_cre,x,2);
" (Christan Stengg on gmane.comp.mathematics.maxima.general)
map(float,1,2,3,4);
float evaluates to false
Improper name or value in functional position.
-- an error. To debug this try debugmode(true);
map('float,1,2,3,4);
[1.0,2.0,3.0,4.0]
Checking precision of floats and bigfloats ( by Barton Willis) :
(%i1) load(floatproperties)$
(%i2) float_eps();
(%o2) 1.1107651257113995*10-16
(%i3) largest_float;
(%o3) 1.7976931348623157*10308
(%i4) least_positive_float;
(%o4) 4.9406564584124654*10-324
(%i5) float_bits();
(%o5) 53
(%i6) bigfloat_eps();
(%o6) 1.387778780781446b-17
(%i7) bigfloat_eps(),fpprec : 100;
(%o7) 1.4287342391028437277697294353[46
digits]3422744594310027325215037b-101
"The e indicator is the normal syntax for floating-point numbers.
's' indicates single-precision floating-point
'd' indicates double-precision floating-point.
I believe all Maxima implementations represent *all* machine floating-point numbers as double-precision, so all three are equivalent.
'b' is Maxima syntax for arbitrary-precision floating point numbers, called 'bigfloats' or 'bfloats'.
The variable fpprec determines how many decimal digits are carried along. Thus:
1/3b0, fpprec=20;
=> 3.3333333333333333333b-1
1/3b0, fpprec=30;
=> 3.33333333333333333333333333333b-1
" ( by Stavros Macrakis )
to draw gradient vectors that are orthogonal to contour lines :
plot2d(x^2,[x,0,3],[plot_format,openmath]);
pplot2d(x^2,[x,0,3],[plot_format,openmath]);
"You should install the latest version of xmaxima that is distributed
with Maxima" Jaime Villate
Plotting the gradient with plotdf
(%i1) load("plotdf.lisp")$
Running Maxima
maxima(1) - Linux man page How to run maxima from console
quit();
"maxima-init.mac is a file which is loaded automatically when Maxima starts.
You can use maxima-init.mac to customize your Maxima environment.
maxima-init.mac, if it exists, is typically placed in the directory named by maxima_userdir, although it can be in any directory searched by the function file_search."
In my Maxima under win XP ther is no maxima-init.mac file.
See Introduction to Maxima by Edwin L. (Ted) Woollett for more informations
(%i2) maxima_userdir;
(%o2) C:/Documents and Settings/adam/maxima
(%i1) file_search_maxima;
in windows :
append(["C:/work2/###.{mac,max}", "C:/work1/temp/###.{mac,max}"], file_search_maxima)$
or in Linux :
append(["/home/adam/###.{mac,dat}"], file_search_maxima)$
then search for file in home dir :
(%i3) file_search_lisp;
file_search("cc1.dat");
printfile("cc1.dat");
cdata : read_list("cc1.dat");
Maxima installation :
Maxima Installation by Mario Rodríguez Riotorto
Getting started in Common Lisp on Ubuntu
SLIME
Maxima installation/compilation Mini-Howto
gcl
Save text file in Maxima :
f : openw (f_name),
printf(f, "~a~%", comment),
printf(f, "~a~%", "dri"),
printf(f, "~d~%", 0),
printf(f, "~d~%", degree),
for x in a do printf (f, "~d~%", x),
close (f));
Save svg file in Maxima :
Under windows it makes an svg file ( for me in C:\Program Files\Maxima-5.21.1\wxMaxima directory)
See also :
Maxima and memory, allocate
Jungreis algorithm
IIM/J in maxima
Main page
Feel free to e-mail me!
Author: Adam Majewski adammaj1-at-o2-dot-pl
http://republika.pl/fraktal/
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