PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Applied Mathematics → Numerical Analysis ↓


Interpolation


Introduction

Interpolation is a fundamental concept in numerical analysis and is widely used in applied mathematics. It involves finding a function that passes through a set of given data points. This function can then be used to estimate or predict values at other points. Simply put, interpolation fills in the gaps between known data points.

Why interpolation?

In many practical situations, we have data collected at discrete points, but we are interested in understanding the behavior of the data as a continuous function. Interpolation helps to construct new data points within the range of a known set of data points.

Applications of interpolation

  • Engineering: To estimate material stresses at unknown points.
  • Weather forecasting: Predicting conditions in surrounding areas using temperature and pressure data from specific locations.
  • Computer graphics: rendering curves and surfaces smoothly.

Basic concept of interpolation

Suppose you have a set of data points:

(x0, y0), (x1, y1), ..., (xn, yn)

Here, each pair represents a known coordinate. The task of interpolation is to find a function f(x) such that:

f(xi) = yi, for i = 0, 1, 2, ..., n

The function f(x) can be used to estimate values at points between x0 and xn.

Types of interpolation

Various methods are used in interpolation, some of which are as follows:

  1. Linear interpolation
  2. Polynomial interpolation
  3. Spline interpolation

Linear interpolation

Linear interpolation is the simplest form. It connects two consecutive data points with a straight line and is suitable when the data are expected to vary linearly.

For two given points (x0, y0) and (x1, y1), the linear interpolation formula is:

f(x) = y0 + (x - x0) * (y1 - y0) / (x1 - x0)

Suppose you have points (1, 2) and (4, 3). The interpolation value at x = 2.5 will be calculated as follows:

f(2.5) = 2 + (2.5 – 1) * (3 – 2) / (4 – 1) = 2 + 1.5 * 1 / 3 ≈ 2.5
(1, 2) (4, 3) (2.5, 2.5)

Polynomial interpolation

Polynomial interpolation involves finding a polynomial of degree n that passes through all n+1 data points. A well-known method of polynomial interpolation is Lagrange interpolation.

Lagrange interpolation

The Lagrange interpolation polynomial f(x) that goes through the points (x0, y0), (x1, y1), ..., (xn, yn) is given by:

f(x) = Σ (yj * Lj(x)) j = 0 to n

where Lj(x) is defined as:

Lj(x) = Π ((x - xi) / (xj - xi)) i = 0 to n, i ≠ j

Let us perform the interpolation for the points (1, 1), (3, 2) and (5, 1) with x = 2:

Calculate L0(x), L1(x), and L2(x):

L0(x) = ((x - 3)(x - 5)) / ((1 - 3)(1 - 5))
L1(x) = ((x - 1)(x - 5)) / ((3 - 1)(3 - 5))
L2(x) = ((x - 1)(x - 3)) / ((5 - 1)(5 - 3))

Thus, f(2) can be calculated using these functions.

(11) (3, 2) (5, 1)

Spline interpolation

Spline interpolation is used when the high degree of the polynomial leads to oscillations between data points (known as Runge's phenomenon). Splines are polynomials divided into pieces that ensure smooth transitions.

The most common spline is the cubic spline, which is composed of cubic polynomials in each interval of the data points and has continuous first and second derivatives.

Suppose you have data points (1, 1), (2, 4), (3, 9). A cubic spline will easily fit a curve through these points. The equations for splines are a little more complicated and involve setting up a system of equations to solve for the coefficients.

S(x) = ai + bi(x - xi) + ci(x - xi)^2 + di(x - xi)^3

Advantages and disadvantages

Benefit

  • Allows to estimate intermediate values.
  • Can help create smooth and continuous curvature.
  • Useful in data fitting and numerical analysis.

Loss

  • High-degree polynomial interpolation can produce oscillations.
  • Complex data may require sophisticated interpolation methods.
  • The greater the number of points, the more complex the interpolation.

Conclusion

Interpolation remains an important technique in numerical analysis and applied mathematics, helping to creatively fill in gaps when clear data is not present. Understanding different interpolation methods allows for more robust data analysis and interpolation that is adaptable to a variety of datasets, ensuring we can make reliable predictions and fits to our data.


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Interpolation

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