Chapter 1: What is Numerical Computation?
What is Numerical Computation?
What is Numerical Computation?
Numerical computation is the method of solving mathematical problems numerically using computers.
In mathematics, it is sometimes possible to express the solution of an equation using a formula or to compute an integral symbolically. However, in many real-world problems, such "clean answers" are rarely obtainable. Numerical computation provides methods for finding "approximate numerical answers" to such problems.
Example: Quadratic vs. Quintic Equations
The solutions of a quadratic equation $ax^2 + bx + c = 0$ can be found using the quadratic formula:
$$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$However, there is no general formula for solving equations of degree five or higher (Abel-Ruffini theorem). Numerical computation is required to solve such equations.
Analytical Solutions vs Numerical Solutions
Example: Solving $x^2 = 2$
Analytical solution (exact solution): $x = \sqrt{2}$ — an exact solution expressed as a formula
Numerical solution (approximate solution): $x \approx 1.41421356237...$ — a concrete number
Analytical solutions are exact, but in practice there are many situations where concrete numerical values are needed. Numerical analysis is the discipline of efficiently finding such numerical solutions.
| Aspect | Analytical Solution | Numerical Solution |
|---|---|---|
| Form | Formulas / Symbols | Concrete numbers |
| Precision | Exact | Approximate (with error) |
| Applicability | Limited | Broad |
| Advantage | Reveals general properties | Provides practical values |
The Concept of Approximation
The core of numerical computation is "approximation." Rather than seeking a perfect answer, the goal is to efficiently find a "sufficiently good answer."
Why Approximation is Necessary
- Many problems cannot be solved analytically
- Computers can only handle a finite number of digits
- In practice, a certain level of precision is sufficient
Example: Approximation of Pi
Pi $\pi = 3.14159265358979...$ is an infinite decimal. However, in practice:
- Elementary school: $\pi \approx 3.14$
- Engineering calculations: $\pi \approx 3.14159$
- High-precision calculations: $\pi \approx 3.141592653589793$ (double precision)
Approximate values are used at a level of precision appropriate to the application.
Applications of Numerical Computation
Solving Equations
Finding $x$ such that $f(x) = 0$. Example: solving $x^5 - x - 1 = 0$
Numerical Integration
Computing the value of $\displaystyle\int_a^b f(x)\,dx$. Example: $\displaystyle\int_0^1 e^{-x^2}\,dx$
Differential Equations
Weather forecasting, fluid simulations, etc.
Matrix Computation
Systems of linear equations, data analysis, machine learning
Importance of Algorithms
In numerical computation, "how the computation is carried out" is extremely important. Even for the same problem, the choice of algorithm affects:
- Speed can vary dramatically
- Accuracy can differ
- Stability (accumulation of errors) can be different
Example: Finding $\sqrt{2}$ with Two Algorithms
The positive solution of $x^2 = 2$ ($\sqrt{2} = 1.41421356237...$) is computed using a slow algorithm and a fast algorithm for comparison.
Method 1: Bisection Method (slow)
Since $1^2 = 1 < 2$ and $2^2 = 4 > 2$, the solution lies in the interval $[1, 2]$. The midpoint of the interval is evaluated, and the interval is narrowed to the half containing the solution. This process is repeated.
| Step | Approximation | Error |
|---|---|---|
| 1 | 1.5 | $8.6 \times 10^{-2}$ |
| 5 | 1.40625 | $8.0 \times 10^{-3}$ |
| 10 | 1.4150390625 | $8.3 \times 10^{-4}$ |
Even after 10 steps, the error is on the order of $10^{-4}$. Precision improves by about 1 bit per step.
Method 2: Newton's Method (fast)
Starting with initial value $x_0 = 1$, the following formula is iterated:
$$x_{n+1} = \dfrac{1}{2}\left(x_n + \dfrac{2}{x_n}\right)$$| Step | Approximation | Error |
|---|---|---|
| 1 | 1.5 | $8.6 \times 10^{-2}$ |
| 2 | 1.41666... | $2.5 \times 10^{-3}$ |
| 3 | 1.41421568... | $2.1 \times 10^{-6}$ |
| 4 | 1.41421356237... | $1.6 \times 10^{-12}$ |
Only 4 steps to reach error $10^{-12}$. The number of correct digits roughly doubles with each step.
Even for the same problem, the convergence speed differs dramatically depending on the choice of algorithm.
Summary
- Numerical computation is the method of solving mathematical problems numerically
- Analytical solutions are formulas; numerical solutions are concrete numbers
- Many problems cannot be solved analytically, so approximation is necessary
- The choice of algorithm affects both accuracy and speed
- Numerical computation is used in various fields of science and engineering
Frequently asked questions
What is numerical computation (numerical analysis)?
Numerical computation is a general term for methods that approximately solve mathematical problems as concrete numerical values that computers can handle. It is used to solve problems such as differential equations, integrals, and systems of equations to practical accuracy when exact analytical solutions cannot be obtained.
What is the difference between analytical solutions and numerical solutions?
An analytical solution is an exact solution expressed in formulas and symbols, such as $\sqrt{2}$ or $\pi$. A numerical solution is an approximate solution expressed as concrete numbers like $1.41421356\ldots$, which contains error but can be used directly in practical calculations.
In what fields is numerical computation used?
It is used across a wide range of science and engineering fields: root-finding, numerical integration, differential equations (weather forecasting, fluid simulation), and matrix computation (systems of equations, data analysis, machine learning). Many problems that cannot be solved analytically, or that require large amounts of computation, rely on numerical computation.
How much does computation speed differ depending on the algorithm?
When finding $\sqrt{2}$, the bisection method achieves an error of about $10^{-4}$ after 10 steps, while Newton's method reaches an error of $10^{-12}$ in only 4 steps. The choice of algorithm has an order-of-magnitude impact on computational efficiency.
Do numerical solutions always contain error?
In most cases, round-off error (from representing values with a finite number of digits) and truncation error (from approximating an infinite process with finitely many steps) arise. However, by choosing an appropriate algorithm, the error can be made small enough for practical purposes.