Welcome to phcpy’s documentation!

Contents:

• Getting Started
  • what is phcpy?
  • installing phcpy
  • extending SageMath with phcpy
  • project history
  • references
  • acknowledgments
• Tutorial
  • A Counter Example to Koushnirenko’s Conjecture
    • solving the system on many cores
    • extracting the real roots
    • plotting the curves
  • A 4-Bar Mechanism
    • solving a polynomial system
    • a straight-line configuration
    • a four-bar mechanism
    • the coupler curve
  • Two Lines Meeting Four Given Lines
    • solving a general instance
    • four real lines
    • visualization
  • The Circle Problem of Apollonius
    • the polynomial systems
    • plotting circles
    • solving polynomial systems
    • a special problem
    • a perturbed problem
  • All Lines Tangent to Four Spheres
    • centers and radii
    • formulating the equations
    • solving the problem
    • the tangent lines
    • plotting the lines
  • Tracking Paths Step by Step
    • plotting solutions paths
    • plotting paths and poles
  • Adjacent 2-by-2 Minors
    • computing a witness set
    • monodromy breakup
  • Design of a moving 7-bar mechanism
    • a Laurent polynomial system
    • a special problem
    • a numerical irreducible decomposition
  • Tangent Lines via Witness Set Intersection
    • a witness set of the circle
    • intersecting with the Jacobian
    • extending with slack variables
    • intersecting two witness sets
    • plotting the tangent lines
• User Manual
  • The Blackbox Solver
    • approximating all isolated solutions
    • options of the solve function
  • Numerical Polynomials
    • symbols
    • numbers
  • Numerical Solutions
    • attributes of numerical solutions
    • filtering solution lists
  • Reproducible Runs
    • solving the cyclic 5-roots system twice
    • fixing the seed
  • Equation and Variable Scaling
    • solving without scaling
    • solving after scaling
  • Parallel Runs
    • speeding up the solver on multiple cores
    • quality up
  • Root Counts and Start Systems
    • total degree
    • multihomogeneous Bezout numbers
    • linear-product start systems
    • mixed volumes
  • Increment-and-Fix Aposteriori Step Control
    • let the path trackers run
    • tuning tolerances of the path trackers
    • a step-by-step path tracker
  • Increment-and-Fix Apriori Step Control
    • let the path trackers run
    • tuning tolerances of the path trackers
    • a step-by-step path tracker
  • Homotopies for Problems in Enumerative Geometry
    • Pieri homotopies
    • Littlewood-Richardson homotopies
  • Newton’s Method, Deflation and Multiplicity
    • Newton’s method
    • deflation
    • multiplicity structure
  • Arc Length Parameter Continuation
    • computing a real quadratic turning point
    • complex parameter homotopy continuation
  • Power Series Expansions
    • Taylor series and Pade approximants
    • expansions starting at series
  • Positive Dimensional Solution Sets
    • witness sets
    • homotopy membership test
    • monodromy breakup
    • cascade of homotopies
    • numerical irreducible decomposition
    • diagonal homotopies
  • Code Snippets
    • the blackbox solver
    • path trackers
    • sweep homotopies
    • Schubert calculus
    • power series expansions
    • positive dimensional solution sets
• Programmer Manual
  • the version module
  • a blackbox solver for isolated solutions
    • functions in the module dimension
    • functions in the module polynomials
    • functions in the module solutions
    • functions in the module volumes
    • functions in the module solver
    • functions in the module examples
    • functions in the module families
  • homotopy methods and path tracking algorithms
    • functions in the module homotopies
    • functions in the module starters
    • functions in the module trackers
    • functions in the module tropisms
    • functions in the module sweepers
    • functions in the module series
    • functions in the module curves
    • functions in the module deflation
  • homotopies for enumerative geometry
    • functions in the module schubert
  • numerical irreducible decomposition
    • functions in the module sets
    • functions in the module cascades
    • functions in the module diagonal
    • functions in the module factor
    • functions in the module decomposition
    • functions in the module binomials
• Indices and tables