Uploaded April 2022 | Updated September 2026, 2 weeks ago
Generalization of the Bisection Method for solving systems of equations. This lesson explains the algorithm for a 2 dimension example based on Harvey-Stenger's approach using bisecting triangles. It includes a visualization of the method in action on an example nonlinear system. Other methods for solving in 3 dimensions and for larger systems are also discussed as well as hybrid approaches. Example code hosted on GitHub github.com/osveliz/numerical-veliz written in Python and using numpy and matplotlib.
Chapters:
0:00 Intro
0:10 Literature
0:47 Bisection Shapes
1:12 Harvey-Stenger
1:29 2D Bisection Setup
2:44 Triangle Point L-Test
3:00 Triangle Point Area-Test
4:24 Picking a Triangle
4:52 Stopping Criteria
5:07 2D Bisection Visualized
5:56 Generalized Bisection Algorithm
7:02 Picking Starting Points
7:38 Harvey-Stenger Algorithm
8:20 Hybrid?
8:50 Oscar's Notes
9:30 Outro
Recommended Viewing:
Bisection Method youtu.be/MlP_W-obuNg
Generalized Secant Method youtu.be/p2OPlnHJPNI
Generalized False Position Method and Alternative Secant Methods youtu.be/c2kSfJ8of7E
Generalized Newton's Method youtu.be/p0SBubUfwiI
Brent's Method youtu.be/-bLSRiokgFk
Newton-Bisection Hybrid youtu.be/FD3BPTMGJds
Reference links:
"A two-dimensional analogue to the method of bisections for solving nonlinear equations" by Charles Harvey and Frank Stenger doi.org/10.1090/S0025-5718-1979-0521286-6
"A three-dimensional analogue to the method of bisections for solving nonlinear equations" by Krzysztof Sikorski doi.org/10.1090/S0025-5718-1979-0521286-6
"A bisection method for systems of nonlinear equations" by Eiger et. al. doi.org/10.1145/2701.2705
"An efficient degree-computation method for a generalized method of bisection" by Baker Kearfott doi.org/10.1007/BF01404868
"Abstract generalized bisection and a cost bound" by Baker Kearfott doi.org/10.1090/S0025-5718-1987-0890261-9
"Solving systems of nonlinear equations using the nonzero value of the topological degree" by Michael N. Vrahatis doi.org/10.1145/50063.214384
Background music "Drifting at 432 Hz" by @UnicornHeads
#NumericalAnalysis #BisectionMethod #NonlinearSystem
Generalization of the Bisection Method for solving systems of equations. This lesson explains the algorithm for a 2 dimension example based on Harvey-Stenger's approach using bisecting triangles. It includes a visualization of the method in action on an example nonlinear system. Other methods for solving in 3 dimensions and for larger systems are also discussed as well as hybrid approaches. Example code hosted on GitHub github.com/osveliz/numerical-veliz written in Python and using numpy and matplotlib.
Chapters:
0:00 Intro
0:10 Literature
0:47 Bisection Shapes
1:12 Harvey-Stenger
1:29 2D Bisection Setup
2:44 Triangle Point L-Test
3:00 Triangle Point Area-Test
4:24 Picking a Triangle
4:52 Stopping Criteria
5:07 2D Bisection Visualized
5:56 Generalized Bisection Algorithm
7:02 Picking Starting Points
7:38 Harvey-Stenger Algorithm
8:20 Hybrid?
8:50 Oscar's Notes
9:30 Outro
Recommended Viewing:
Bisection Method youtu.be/MlP_W-obuNg
Generalized Secant Method youtu.be/p2OPlnHJPNI
Generalized False Position Method and Alternative Secant Methods youtu.be/c2kSfJ8of7E
Generalized Newton's Method youtu.be/p0SBubUfwiI
Brent's Method youtu.be/-bLSRiokgFk
Newton-Bisection Hybrid youtu.be/FD3BPTMGJds
Reference links:
"A two-dimensional analogue to the method of bisections for solving nonlinear equations" by Charles Harvey and Frank Stenger doi.org/10.1090/S0025-5718-1979-0521286-6
"A three-dimensional analogue to the method of bisections for solving nonlinear equations" by Krzysztof Sikorski doi.org/10.1090/S0025-5718-1979-0521286-6
"A bisection method for systems of nonlinear equations" by Eiger et. al. doi.org/10.1145/2701.2705
"An efficient degree-computation method for a generalized method of bisection" by Baker Kearfott doi.org/10.1007/BF01404868
"Abstract generalized bisection and a cost bound" by Baker Kearfott doi.org/10.1090/S0025-5718-1987-0890261-9
"Solving systems of nonlinear equations using the nonzero value of the topological degree" by Michael N. Vrahatis doi.org/10.1145/50063.214384
Background music "Drifting at 432 Hz" by @UnicornHeads
#NumericalAnalysis #BisectionMethod #NonlinearSystem
