bairstow.hpp file
Typedefs
-
using Vec2 = ginger::
Vector2<double> -
using Mat2 = ginger::
Matrix2<Vec2> -
using Vec2Ref = ginger::
Vector2<double&>
Functions
- auto horner_ref(std::vector<double>& coeffs, std::vector<Vec2Ref>& vcoeffs, std::size_t degree, const Vec2& vr) -> Vec2Ref -> auto
- Horner's rule.
- auto bairstow(const std::vector<double>& coeffs, Vec2& vr, const Options& options) -> std::pair< unsigned int, bool > -> auto
- Bairstow's method (even degree only)
- auto makeadjoint_ref(const Vec2& vr, const Vec2Ref& vp) -> Mat2 -> auto
- auto delta_ref(const Vec2Ref& vA, const Vec2& vr, const Vec2Ref& vp) -> Vec2 -> auto
Function documentation
auto horner_ref(std::vector<double>& coeffs, std::vector<Vec2Ref>& vcoeffs, std::size_t degree, const Vec2& vr) -> Vec2Ref
Horner's rule.
| Parameters | |
|---|---|
| coeffs | |
| vcoeffs | |
| degree in | The parameter degree represents the size of the vector coeffs1. It indicates the number of elements in the vector coeffs1. |
| vr in | vr is a Vec2 object, which represents a 2D vector. It has two components, vr.x() and vr.y(), which are used in the calculations inside the horner function. |
| Returns | a Vec2 object. |
Horner's rule is a method for evaluating a polynomial of degree degree at a given point x. It involves rewriting the polynomial as a nested multiplication and addition of the form:
P(x) = a_0 + x(a_1 + x(a_2 + ... + x(a_{degree-1} + x(a_n))...))
This form allows for efficient evaluation of the polynomial at a given point x using only degree multiplications and degree additions. Horner's rule is commonly used in numerical methods for polynomial evaluation and interpolation.
auto bairstow(const std::vector<double>& coeffs, Vec2& vr, const Options& options) -> std::pair< unsigned int, bool >
Bairstow's method (even degree only)
| Parameters | |
|---|---|
| coeffs in | The coeffs parameter is a vector representing the coefficients of the polynomial. Each element of the vector corresponds to the coefficient of a term in the polynomial, starting from the highest degree term and ending with the constant term. For example, if the polynomial is 3x^2 + 2 @param[in, out] vrvris a vector of iterates, which represents the initial guesses for the roots of the polynomial. The Bairstow's method will update these iterates iteratively until the desired tolerance is reached or the maximum number of iterations is reached. @param[in] options Theoptionsparameter is an object of typeOptionswhich contains the maximum number of iterations (max_iters) and the tolerance (tolerance`). These options are used to control the convergence criteria for the Bairstow's method. |
| vr | |
| options | |
| Returns | The function pbairstow_even returns a std::pair<unsigned int, bool>. The first element of the pair represents the number of iterations performed, and the second element represents whether the method converged to a solution within the specified tolerance. |
The bairstow_even function implements Bairstow's method for finding the roots of a real polynomial with an even degree using multi-threading.
auto makeadjoint_ref(const Vec2& vr, const Vec2Ref& vp) -> Mat2
| Parameters | |
|---|---|
| vr in | A constant reference to a Vec2 object, representing the vector vr. |
| vp in | vp is a vector with two components, vp.x() and vp.y(). |
| Returns | a Mat2 object. |
The function "makeadjoint" takes in a vector vr and a vector vp, and returns a 2x2 matrix where the elements are calculated based on the values of vr and vp.
auto delta_ref(const Vec2Ref& vA, const Vec2& vr, const Vec2Ref& vp) -> Vec2
| Parameters | |
|---|---|
| vA in | A vector of type Vec2. |
| vr in | A vector representing the direction of rotation. |
| vp in | The parameter vp is a Vec2 object that is passed by rvalue reference. |
| Returns | a Vec2 object. |
The function calculates the delta value using the given parameters.