Numerical Analysis. Professor Donna Calhoun. Fall 2013 Math 465/565. Office : MG241A Office Hours : Wednesday 10:00-12:00 and 1:00-3:00

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Numerical Analysis

Professor Donna Calhoun

Office : MG241A

Office Hours : Wednesday 10:00-12:00 and 1:00-3:00

Fall 2013

Math 465/565

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What is Numerical Analysis

Numerical analysis is a branch of mathematics that deals with the understanding of how algorithms designed to

solve mathematically posed questions behave when carried out (mostly on a computer).

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Consistency (does our algorithm approximate our

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mathematically posed question in some way?

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Stability (is our algorithm well-behaved in some sense

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and likely to run to completion?)

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Convergence (do we approach the right answer as we

increase the time or computational resources we allocate to a problem?)

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and likely to run to completion?)

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Convergence (do we approach the right answer as we

increase the time or computational resources we allocate to a problem?)

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Scalability (can we scale our algorithm to large versions

of our problem without requiring exponentially more resources?)

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Numerical analysis and computational

science

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Numerical analysis and computational

science

Sidebar 1

Definition of Computational Science

As a basis for responding to the charge from the Office of Science and

Technology Policy, the PITAC developed a definition of computational science. This definition recognizes the diverse components, ranging from algorithms, software, architecture, applications, and infrastructure that collectively represent

computational science.

Computational science is a rapidly growing multidisciplinary field that uses advanced computing capabilities to understand and solve complex problems. Computational science fuses three distinct elements:

• Algorithms (numerical and non-numerical) and modeling and simulation

software developed to solve science (e.g., biological, physical, and social), engineering, and humanities problems

• Computer and information science that develops and optimizes the advanced system hardware, software, networking, and data management components needed to solve computationally demanding problems

• The computing infrastructure that supports both the science and engineering problem solving and the developmental computer and information science

P R E S I D E N T ’ S I N F O R M AT I O N T E C H N O L O G Y A D V I S O R Y C O M M I T T E E

of disease, atom-by-atom construction of materials with previously

unimaginable properties, miniaturization of devices down to the quantum level, and new energy sources and fuel technologies. But we are not minding the store of U.S. intellectual resources needed to capitalize on the scientific opportunities of the new century.

A dangerous consequence of our current complacency is that, as on the eve of Sputnik’s launch, we have not marshaled and focused our efforts to elevate computational science and the computing infrastructure to their appropriate status as a long-term, strategic national priority in education as well as R&D. Without such a commitment and focus, the PITAC believes, we cannot

sustain U.S. scientific leadership, security, and economic prosperity in the decades ahead.

What Is Computational Science?

At one level, computational science is simply the application of computing capabilities to the solution of problems in the real world – for example,

enabling biomedical researchers rapidly to identify to which protein, and where on that protein, a candidate vaccine will most effectively bind. The PITAC’s definition of computational science (Sidebar 1, below, and Figure 1 on page 11) is intended, however, to underscore the reality that harnessing software, hardware, data, and connectivity to help solve complex problems necessarily draws on the multidisciplinary skills represented in the computing infrastructure as a whole.

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