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Numerical Analysis
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1. Interpolation
- Piecewise Interpolation
- Lagrange Interpolation: Equal Spacing
- Lagrange Polynomial: Even/Odd Sums
- Cubic Spline
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2. Example: Adpating A Bezier Curve
- Bezier Approximation, 1
- Bezier Approximation, 2
- Bezier Circle
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3. Quadrature, 1
- Quadrature: Approximate Osculating Circles
- The Arc Tangent
- Bezier Curve Arc Length: Polynomial Approximation
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4. Zeros of a Polynomial
- Newton's Method
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5. The Cubic Equation
- Cubic Equation, 1
- Cubic Equation: bounds
- Roots of the Cubic Equation: Solution
- Roots of the Cubic Equation: Solution 1 (Iteration)
- Quadratic approximation
- One root as a continued fraction
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6. Trigonometric sums
- Approximate the function cos(x)
- Trigonometric Polynomial, 1
- Trigonometric Polynomial, 2
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7. Tools
- Tool: Cubic Spline (Construction)
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Numerical Analysis
Ryan Hirst, Aug 13, 2014
Part of a project to develop a solution to the general problem f(x) =0. Following roughly the order of Ralston's A First Course In Numercal Analysis, worksheets examining the related problems of Interpolation Numerical differentiation and quadrature Numerical solution of differential equations The zeros of a polynomial Approximation error
Table of Contents
- Interpolation
- Piecewise Interpolation
- Lagrange Interpolation: Equal Spacing
- Lagrange Polynomial: Even/Odd Sums
- Cubic Spline
- Example: Adpating A Bezier Curve
- Bezier Approximation, 1
- Bezier Approximation, 2
- Bezier Circle
- Quadrature, 1
- Quadrature: Approximate Osculating Circles
- The Arc Tangent
- Bezier Curve Arc Length: Polynomial Approximation
- Zeros of a Polynomial
- Newton's Method
- The Cubic Equation
- Cubic Equation, 1
- Cubic Equation: bounds
- Roots of the Cubic Equation: Solution
- Roots of the Cubic Equation: Solution 1 (Iteration)
- Quadratic approximation
- One root as a continued fraction
- Trigonometric sums
- Approximate the function cos(x)
- Trigonometric Polynomial, 1
- Trigonometric Polynomial, 2
- Tools
- Tool: Cubic Spline (Construction)
Piecewise Interpolation
Piecewise Interpolation, using f(x), f'(x) at the two nearest tabular points.
Piecewise Interpolation
Bezier Approximation, 1
An order n Bezier curve (a vector function) can represent smooth curves not possible with an order n polynomial. Are they a practical choice for function approximation? Consider the order 3 Bezier curve, defined by two points and the tangent vectors at those points:
Bezier Approximation, 1
Quadrature: Approximate Osculating Circles
Before getting into the textbook, let me consider some ways that one might approximate the arc length of a curve.
Firstly, I can always use linear approximation: lay out points along the curve, draw straight lines between adjacent points, and add up the distances. But perhaps I can do better than this.
The lenght of the circular arc through three consecutive points on the curve responds to both first and second derivatives, and should be a much better approximation. Let me try it.
Quadrature: Approximate Osculating Circles
Newton's Method
Consider a sequence of x-values:
Begin with , chosen at pleasure.
Draw the tangent to the curve at x0:
The x-intercept of this tangent line, is the next number in the sequence:
Newton's Method
Cubic Equation, 1
Working through the problem of solving the cubic, the question arises, suppose I remove the offset? That is, I have
which is the same same as
...shifted to the right or left by some unknown number of units.
Writing , the first root is 0, and the last two roots are given by the quadratic equation.
If we write g(x) = f(x-t), is it possible to solve for the mystery offset t and the new mystery coefficients?
Cubic Equation, 1
Approximate the function cos(x)
NOTES:
- The red arc is the 2nd order Bezier curve: p(t)= A + 2(B-A)t + (C-2B+A)t², 0≤t≤1. To find the vertex, focus, and parabola equation: http://www.geogebratube.org/material/show/id/37814
- To convert this to an equation in x: Solve for t. Then The conversion can be put off as long as possible. A, B,C, are sufficient to retrieve both the original function and calculate an approximation.
- The analysis graphs don't prove the results. However, it should be clear that we only need to establish Does the parabola arc cross cos(x)? in the neighborhood of a point where they are tangent.
Tool: Cubic Spline (Construction)
Spline[<Points>, n, f'(a), f'(b)]
gives the cubic spline passing through n equally spaced points, and with slopes f'(a), f'(b) at the endpoints.
Tool: Cubic Spline (Construction)
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