Scientific computing in the context of Sparse matrix

Data science is an interdisciplinary academic field that uses statistics, scientific computing, scientific methods, processing, scientific visualization, algorithms and systems to extract or extrapolate knowledge from potentially noisy, structured, or unstructured data.

Data science also integrates domain knowledge from the underlying application domain (e.g., natural sciences, information technology, and medicine). Data science is multifaceted and can be described as a science, a research paradigm, a research method, a discipline, a workflow, and a profession.

In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects. Although computer algebra could be considered a subfield of scientific computing, they are generally considered as distinct fields because scientific computing is usually based on numerical computation with approximate floating point numbers, while symbolic computation emphasizes exact computation with expressions containing variables that have no given value and are manipulated as symbols.

Software applications that perform symbolic calculations are called computer algebra systems, with the term system alluding to the complexity of the main applications that include, at least, a method to represent mathematical data in a computer, a user programming language (usually different from the language used for the implementation), a dedicated memory manager, a user interface for the input/output of mathematical expressions, and a large set of routines to perform usual operations, like simplification of expressions, differentiation using the chain rule, polynomial factorization, indefinite integration, etc.

A computational scientist is a person skilled in scientific computing. This person is usually a scientist, a statistician, an applied mathematician, or an engineer who applies high-performance computing and sometimes cloud computing in different ways to advance the state-of-the-art in their respective applied discipline; physics, chemistry, social sciences and so forth. Thus scientific computing has increasingly influenced many areas such as economics, biology, law, and medicine to name a few. Because a computational scientist's work is generally applied to science and other disciplines, they are not necessarily trained in computer science specifically, though concepts of computer science are often used. Computational scientists are typically researchers at academic universities, national labs, or tech companies.

One of the tasks of a computational scientist is to analyze large amounts of data, often from astrophysics or related fields, as these can often generate huge amounts of data. Computational scientists often have to clean up and calibrate the data to a usable form for an effective analysis. Computational scientists are also tasked with creating artificial data through computer models and simulations.

Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain differential equations. The idea is to write the solution of the differential equation as a sum of certain "basis functions" (for example, as a Fourier series which is a sum of sinusoids) and then to choose the coefficients in the sum in order to satisfy the differential equation as well as possible.

Spectral methods and finite-element methods are closely related and built on the same ideas; the main difference between them is that spectral methods use basis functions that are generally nonzero over the whole domain, while finite element methods use basis functions that are nonzero only on small subdomains (compact support). Consequently, spectral methods connect variables globally while finite elements do so locally. Partially for this reason, spectral methods have excellent error properties, with the so-called "exponential convergence" being the fastest possible, when the solution is smooth. However, there are no known three-dimensional single-domain spectral shock capturing results (shock waves are not smooth). In the finite-element community, a method where the degree of the elements is very high or increases as the grid parameter h increases is sometimes called a spectral-element method.