MATLAB Examples

Intersection of circle with y axis

Example 1.9 from Numerically solving polynomial systems with Bertini, by Daniel J. Bates, Jonathan D. Haunstein, Andrew J. Sommese and Charles W. Wampler (SIAM 2013).

Find the zeros of

$$f(x,y) = \left((x-c)^2+y^2-1,x\right)$$

for $c=1$ (which produces a singular solution). The constant $c$ can be modified to move the circle left or right.

poly_system = BertiniLab('function_def',{'(x-c)^2+y^2-1';'x'},'variable_group',{'x','y'}, ...
    'constant',{'c',1});
poly_system = poly_system.solve;

The summary for this run is:

disp(poly_system.solve_summary)
    Bertini(TM) v1.4
   (October 23, 2013)

 D.J. Bates, J.D. Hauenstein,
 A.J. Sommese, C.W. Wampler

(using GMP v5.1.3, MPFR v3.1.2)



NOTE: You have requested to use adaptive path tracking.  Please make sure that you have
setup the following tolerances appropriately:
CoeffBound: 2.736856000000e+00, DegreeBound: 2.000000000000e+00
AMPSafetyDigits1: 1, AMPSafetyDigits2: 1, AMPMaxPrec: 1024

Tracking path 0 of 2

Finite Solution Summary

NOTE: nonsingular vs singular is based on condition number and identical endpoints

		| Number of real solns	|  Number of non-real solns	|  Total
------------------------------------------------------------------------------------------
Non-singular	|	0		|		0		|   0
Singular	|	1		|		0		|   1
------------------------------------------------------------------------------------------
Total		|	1		|		0		|   1


Finite Multiplicity Summary

  Multiplicity	|  Number of real solns	|  Number of non-real solns
------------------------------------------------------------------------------------------
	2	|	1		|	0


------------------------------------------------------------------------------------------
The following files may be of interest to you:

main_data:             A human-readable version of the solutions - main output file.
raw_solutions:         A list of the solutions with the corresponding path numbers.
raw_data:              Similar to the previous, but with the points in Bertini's homogeneous
                         coordinates along with more information about the solutions.
real_finite_solutions: A list of all real finite solutions.
finite_solutions:      A list of all finite solutions.
nonsingular_solutions: A list of all nonsingular solutions.
singular_solutions:    A list of all singular solutions.
------------------------------------------------------------------------------------------

Paths Tracked: 2

The solutions are singular, being nearly identical and having large condition numbers. The final tolerance by default is $10^{-11}$; displayed to that accuracy, the roots are identical:

sols = poly_system.match_solutions('singular_solutions');
solution_info = poly_system.read_raw_data;
condnum = [solution_info.condition_number].';
xsols = double(sols.x); ysols = double(sols.y);
fprintf('%17s %35s %32s\n','x','y','condition number')
fprintf('%15.11f + %15.11fi  %15.11f + %15.11fi %15.2g\n', ...
    [real(xsols) imag(xsols) real(ysols) imag(ysols) condnum].')
                x                                   y                 condition number
 -0.00000000000 +   0.00000000000i    0.00000000000 +   0.00000000000i         1.2e+13
 -0.00000000000 +   0.00000000000i    0.00000000000 +   0.00000000000i         1.2e+13