Numerical Methods I
MATH-GA 2010.001/CSCI-GA 2420.001
Georg Stadler
Courant Institute, NYU
stadler@cims.nyu.edu
Fall 2015, Thursday, 5:10–7:00PM, WWH #101
Sep. 3, 2015
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Outline
Organization issues
Introduction and examples
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Organization issues
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Time and location: Thursday 5:10–7:00PM, WWH 101
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Organization issues
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Time and location: Thursday 5:10–7:00PM, WWH 101
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Course webpage: http://cims.nyu.edu/~stadler/num1/
3 / 14
Organization issues
I
Time and location: Thursday 5:10–7:00PM, WWH 101
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Course webpage: http://cims.nyu.edu/~stadler/num1/
We will use Piazza for communication in this class. Let me
know if you want to be added (per email or in today’s break).
I
3 / 14
Organization issues
I
Time and location: Thursday 5:10–7:00PM, WWH 101
I
Course webpage: http://cims.nyu.edu/~stadler/num1/
We will use Piazza for communication in this class. Let me
know if you want to be added (per email or in today’s break).
I
I
Email: If you email me (stadler@cims.nyu.edu) about
anything related to this course, please put [num1] in the
email’s subject line.
3 / 14
Organization issues
I
Time and location: Thursday 5:10–7:00PM, WWH 101
I
Course webpage: http://cims.nyu.edu/~stadler/num1/
We will use Piazza for communication in this class. Let me
know if you want to be added (per email or in today’s break).
I
I
Email: If you email me (stadler@cims.nyu.edu) about
anything related to this course, please put [num1] in the
email’s subject line.
I
Office hours: Wednesday, 10AM-noon, stop by or make an
appointment (please email). My office number is #1111.
3 / 14
Organization issues
I
Time and location: Thursday 5:10–7:00PM, WWH 101
I
Course webpage: http://cims.nyu.edu/~stadler/num1/
We will use Piazza for communication in this class. Let me
know if you want to be added (per email or in today’s break).
I
I
Email: If you email me (stadler@cims.nyu.edu) about
anything related to this course, please put [num1] in the
email’s subject line.
I
Office hours: Wednesday, 10AM-noon, stop by or make an
appointment (please email). My office number is #1111.
I
Survey: I will hand out a survey today that will help me get to
know your backgrounds/experience/interests better.
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Organization issues
Prerequisites:
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Basic linear algebra; calculus; some experience in Matlab (or
another programming language)
There is a part II of this class. . .
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. . . in spring 2016 (taught by Jonathan Goodman). You
should take both parts to get a completer picture of
Numerical Methods.
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If you only consider taking one part of Numerical
Mathematics, consider taking Scientific Computing (taught by
Aleks Donev) rather than this class.
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Organization issues
Topics of Numerical Methods I
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Stability; sources of errors; error propagation
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Numerical linear algebra: direct solution of sparse/dense linear
system; solution of least square systems; eigenvalue problems;
iterative solution of linear systems (?)
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Nonlinear systems; Newton’s method
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Interpolation and Approximation
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Numerical integration
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Numerical optimization (?)
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MCMC sampling methods (?)
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Organization issues
Topics of Numerical Methods I
I
Stability; sources of errors; error propagation
I
Numerical linear algebra: direct solution of sparse/dense linear
system; solution of least square systems; eigenvalue problems;
iterative solution of linear systems (?)
I
Nonlinear systems; Newton’s method
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Interpolation and Approximation
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Numerical integration
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Numerical optimization (?)
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MCMC sampling methods (?)
Topics of Numerical Methods II
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Approximation of ordinary differential equations (ODEs)
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Approximation of partial differential equations (PDEs)
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Solvers for the resulting discrete problems
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Organization issues
Programming
Matlab:
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Make sure you have access (CIMS, student license), you will
need it for the first homework assignment.
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Alternatives: Octave or Python.
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We will talk about best coding practices, and how to best
present results.
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Compiled languages: I will encourage you to play with C; this
is voluntarily (likely there will be a few extra credit homework
problems)
Programming the methods we discuss is an important part of this
course. To really understand a(n) method/algorithm, one needs to
implement it.
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Organization issues
Recommended textbooks/literature:
Main text books:
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A. Quarteroni, R. Sacco, F. Saleri: Numerical Mathematics,
2nd edition, Springer, 2007.
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P. Deuflhard, A. Hohmann: Numerical Analysis in Modern
Scientific Computing. An Introduction, 2nd edition, Springer,
2003.
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Organization issues
Recommended textbooks/literature:
Main text books:
I
A. Quarteroni, R. Sacco, F. Saleri: Numerical Mathematics,
2nd edition, Springer, 2007.
I
P. Deuflhard, A. Hohmann: Numerical Analysis in Modern
Scientific Computing. An Introduction, 2nd edition, Springer,
2003.
Further reading:
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L. N. Trefethen, D. Bao: Numerical Linear Algebra, SIAM,
1997.
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Organization issues
Recommended textbooks/literature:
Main text books:
I
A. Quarteroni, R. Sacco, F. Saleri: Numerical Mathematics,
2nd edition, Springer, 2007.
I
P. Deuflhard, A. Hohmann: Numerical Analysis in Modern
Scientific Computing. An Introduction, 2nd edition, Springer,
2003.
Further reading:
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L. N. Trefethen, D. Bao: Numerical Linear Algebra, SIAM,
1997.
Matlab/Programming:
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W. Gander, M. J. Gander, F. Kwok: Scientific Computing An Introduction Using Maple and MATLAB. Texts in
Computation Science and Engineering. Springer, 2014.
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C. Moler: Numerical Computing with Matlab, SIAM, 2007.
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Organization issues
Required work and grading:
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Weekly (obligatory and recommended) reading/self-study
assignments.
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Organization issues
Required work and grading:
I
I
Weekly (obligatory and recommended) reading/self-study
assignments.
7 homework assignments (∼70% of your grade).
8 / 14
Organization issues
Required work and grading:
I
I
Weekly (obligatory and recommended) reading/self-study
assignments.
7 homework assignments (∼70% of your grade).
I
These will be mixed paper&pencil and
computational/programming. You hand in solutions (I’ll follow
up with details on format and presentation).
8 / 14
Organization issues
Required work and grading:
I
I
Weekly (obligatory and recommended) reading/self-study
assignments.
7 homework assignments (∼70% of your grade).
I
I
These will be mixed paper&pencil and
computational/programming. You hand in solutions (I’ll follow
up with details on format and presentation).
You are welcome to discuss with your colleagues (and post
issues you find on Piazza), but you’ve to write up your solution
independently and write every line of code yourself.
8 / 14
Organization issues
Required work and grading:
I
I
Weekly (obligatory and recommended) reading/self-study
assignments.
7 homework assignments (∼70% of your grade).
I
I
I
These will be mixed paper&pencil and
computational/programming. You hand in solutions (I’ll follow
up with details on format and presentation).
You are welcome to discuss with your colleagues (and post
issues you find on Piazza), but you’ve to write up your solution
independently and write every line of code yourself.
Please read NYU’s policy on Academic Integrity.
8 / 14
Organization issues
Required work and grading:
I
I
Weekly (obligatory and recommended) reading/self-study
assignments.
7 homework assignments (∼70% of your grade).
I
I
I
I
These will be mixed paper&pencil and
computational/programming. You hand in solutions (I’ll follow
up with details on format and presentation).
You are welcome to discuss with your colleagues (and post
issues you find on Piazza), but you’ve to write up your solution
independently and write every line of code yourself.
Please read NYU’s policy on Academic Integrity.
The first homework assignment will be posted tomorrow.
8 / 14
Organization issues
Required work and grading:
I
I
Weekly (obligatory and recommended) reading/self-study
assignments.
7 homework assignments (∼70% of your grade).
I
I
I
I
I
These will be mixed paper&pencil and
computational/programming. You hand in solutions (I’ll follow
up with details on format and presentation).
You are welcome to discuss with your colleagues (and post
issues you find on Piazza), but you’ve to write up your solution
independently and write every line of code yourself.
Please read NYU’s policy on Academic Integrity.
The first homework assignment will be posted tomorrow.
A take-home final (∼30% of grade).
8 / 14
Organization issues
Required work and grading:
I
I
Weekly (obligatory and recommended) reading/self-study
assignments.
7 homework assignments (∼70% of your grade).
I
I
I
I
I
These will be mixed paper&pencil and
computational/programming. You hand in solutions (I’ll follow
up with details on format and presentation).
You are welcome to discuss with your colleagues (and post
issues you find on Piazza), but you’ve to write up your solution
independently and write every line of code yourself.
Please read NYU’s policy on Academic Integrity.
The first homework assignment will be posted tomorrow.
A take-home final (∼30% of grade).
I
Posted during last class; you will have 5-7 days to work out
problems that slightly extend the topics of the course.
8 / 14
Organization issues
Required work and grading:
I
I
Weekly (obligatory and recommended) reading/self-study
assignments.
7 homework assignments (∼70% of your grade).
I
I
I
I
I
These will be mixed paper&pencil and
computational/programming. You hand in solutions (I’ll follow
up with details on format and presentation).
You are welcome to discuss with your colleagues (and post
issues you find on Piazza), but you’ve to write up your solution
independently and write every line of code yourself.
Please read NYU’s policy on Academic Integrity.
The first homework assignment will be posted tomorrow.
A take-home final (∼30% of grade).
I
I
Posted during last class; you will have 5-7 days to work out
problems that slightly extend the topics of the course.
Absolute no collaboration for the take-home exam.
8 / 14
Organization issues
Summary of resources:
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Books and homework assignments. I’ll also make my slides
available.
I
Piazza–central communication/discussion/announement
platform
I
Smile! These lectures are being video recorded; videos will be
available through NYU Classes.
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Public class website:
http://cims.nyu.edu/~stadler/num1/
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Outline
Organization issues
Introduction and examples
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Numerical mathematics
Computer simulations have had a huge influence on research and
development; sometimes the ability to simulate phenomena is
referred to as the third pillar or science.
Numerical mathematics is a
part of mathematics that
develops, analyzes and applies
methods from scientific
computing to
It has applications accross
many applied sciences,
including:
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physics
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analysis
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economics
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linear algebra
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biology
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optimization
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finance
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differential equations
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...
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...
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Famous numerical mathematics mistakes
Patriot Missile Failure
In the 1991 Gold war, a patriot
missle failed to intercept an Iraqi
Scud missile.
28 US soulders died, 100 were
injured.
Cause: Inaccurate calculation of
the time since boot due to
computer arithmetic errors
http://www.ima.umn.edu/~arnold/disasters/patriot.html
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Famous numerical mathematics mistakes
Sinking of Sleipner oil platform
An oil platform in the North Sea sank
near Stavanger (Norway) in 1991. Top
part weights 57,000 tons, supposed to
support drilling equipment that weights
40,000 tons.
Total economic loss was about 700
million USD.
Cause: Weak parts in the base could
not resist the weight. Stresses were
underestimated by 47%, leading to
insufficient design. This was mainly due
to an inaccurate finite element
calculation to solve the PDE.
http://www.ima.umn.edu/~arnold/disasters/sleipner.html
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Famous numerical mathematics mistakes
Explosion of Ariane 5
Unmanned Ariane 5 rocket
launched by the European space
agency expolded in 1996.
Rocket value was abot 500
million USD.
Cause: Conversion of a floating
point number to an integer led to
“overflow” resulting in complete
loss of guidance and altitude
information 37 seconds after
start.
http://www.ima.umn.edu/~arnold/disasters/ariane.html
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