Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.
Before modern computers, numerical methods often relied on hand interpolation formulas, using data from large printed tables. Since the mid 20th century, computers calculate the required functions instead, but many of the same formulas continue to be used in software algorithms.
The numerical point of view goes back to the earliest mathematical writings. A tablet from the Yale Babylonian Collection (YBC 7289), gives a sexagesimal numerical approximation of the square root of 2, the length of the diagonal in a unit square.
Numerical analysis continues this long tradition: rather than giving exact symbolic answers translated into digits and applicable only to real-world measurements, approximate solutions within specified error bounds are used.
Key aspects of numerical analysis include:
1. Error Analysis: Understanding and minimizing the errors that arise in numerical calculations, such as round-off errors, truncation errors, and approximation errors.
2. Convergence: Determining whether a numerical method will converge to the correct solution as more iterations or finer steps are taken.
3. Stability: Ensuring that small changes in the input or intermediate steps do not cause large changes in the output, which could lead to incorrect results.
4. Efficiency: Developing algorithms that solve problems in a reasonable amount of time and with manageable computational resources.
5. Conditioning: Analyzing how the solution to a problem is affected by small changes in the input data, which helps in assessing the reliability of the numerical solution.
Numerical analysis plays a crucial role in scientific computing, engineering simulations, financial modeling, and many other fields where mathematical modeling is essential.
I'm wondering whether a differential equation that can be integrated numerically forwards in time can also be integrated backwards in time starting from the final state and inverting the momenta/velocities? I tried and it didn't work. But I'm not sure whether I'm making a mistake with my solver...