kill(All)$

appendix : intersection between two function , version 1.0
peter.vlasschaert@gmail.com, 13/12/2020

use reason : eigenvalues .

load packages ,option 
=================

load(lrats)$ ratprint:false$
load(mnewton)$load(newton1)$
load(to_poly_solve)$

first curve 
--------------

p1a:tan(x);

second curve
------------------

p1b:1/x;

intersection(p1a,p1b)
----------------------------

p1:p1a-p1b;

listofvars(p1);

reason : multiple intersection 

wxplot2d([p1a,p1b,p1], [x,0,20],[y,0,2],grid2d)$

find : limit ( = 1/x)

p2:'limit(p1b,x,inf) = limit(p1b,x,inf);

detail :

wxplot2d([p1a,p1b,p1,p1^2], [x,0,1.5],[y,0,2],grid2d)$

 figure: small ranges ( = zoom)
-------------------------------------------

wxplot2d([p1a,p1b,p1,p1^2], [x,0.8,1.1],[y,0.8,1.1],grid2d)$

lambda values : λi ,i=1,2,3,4,....
============

λ1:(newton( tan(x)-1/x,x,0.5,1e-6))^2;

p2a:(newton( tan(x)-1/x,x,0.5,1e-6))^2;/* load newton */

p2b:(find_root(tan(x)-1/x, x, 0.01, 1))^2;

→ check : numerical ( = zero)

tan(0.8603336516072181)-1/(0.8603336516072181);

λ2:(newton( tan(x)-1/x,x,6*p2a,1e-6))^2;

→ check : numerical ( = zero)

tan(6.437298629059229)-1/(6.437298629059229);

λ3:(newton( tan(x)-1/x,x,10*p2a,1e-6))^2;

→ check : numerical ( = zero)

tan(9.52933440550345)-1/(9.52933440550345);

λ4:(newton( tan(x)-1/x,x,13*p2a,1e-6))^2;

rem :
====

→ to_poly_solve?

to_poly_solve([tan(x)-1/x=0], [x]);

→ mnewton       ?

Function: mnewton (FuncList,VarList,GuessList)

   (%i1) load("mnewton")$
   (%i2) mnewton([x1+3*log(x1)-x2^2, 2*x1^2-x1*x2-5*x1+1],
              [x1, x2], [5, 5]);
   (%o2) [[x1 = 3.756834008012769, x2 = 2.779849592817897]]
   (%i3) mnewton([2*a^a-5],[a],[1]);
   (%o3)             [[a = 1.70927556786144]]
   (%i4) mnewton([2*3^u-v/u-5, u+2^v-4], [u, v], [2, 2]);
   (%o4) [[u = 1.066618389595407, v = 1.552564766841786]]

q1a:mnewton([p1],[x],[0.5]);
              

→x^2

q1a1:q1a[1];

→ x^2

q1a2:rhs(q1a1[1])*rhs(q1a1[1]);

q1b:flatten((q1a)^2);

q1b:q1b[1];

check : q1a2 & q1b
---------------------------

is(q1a2=rhs(q1b));