SSG 314 Lists & Other Structures

7/31/2019 SSG 314 Lists & Other Structures

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Lists & Other StructuresSets, vectors, matrices and tensors


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Engineering students at this level are familiar with

vectors and matrices. From a computational point of view, these are

different from scalars in two ways:

They contain more than one scalar value

They are organized in a way that they can be stored andacted upon in specific ways.

Formally, we say these are structured data.

Structuring Data


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Other structures occur in tensor calculus, set theory,

graph theory, etc. In many cases, these structures behave in peculiar

ways. For computational purposes, these areencapsulated in descriptive algorithms that can be

bound with the data. Abstract Data types such as classes in OOP can be

used to create these bindings as we shall see later.

Other Structures


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In Mathematica, the fundamental data structure is

the list. The list is already endued with basic algorithms for

processing.

Other structures can be built with additional

processing algorithms.

List as the fundamental structure


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Generating Lists


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These, in addition to some of the simple matrix

examples we have seen earlier are generated lists. There are other list-creating functions such as Array

and other more verbose iterative methods as you cansee from the cookbook.

In addition to these, lists can be amended byfunctions created for that purpose.

List Generation


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Amending Lists


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Other commands are used to analyse and take listsapart. Some of these commands are used to extractparts of lists:

Part, First, Last, Rest, Most, Take, Drop, Select, CasesReverse, RotateLeft, RotateRight, Flatten, Partition,Transpose, and Sort

Consider Take and Drop:

Analysing Lists


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In Mathematica, A set is a list. In conformity with the

mathematical definition of a set, the elements areunique so that duplicates are eleminated when setoperations such as Union, Intersection or Complimentare applied to them.

You can also check if a particular element belongs to aset. The set itself can be a set of scalars or of lists.

Sets


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There are several set operations that can be

performed by in-built functions. In the examplesbelow, we generate all the subsets of a particular setand later check if a specific element belongs to aparticular defined set:

Set Operations


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A Tensor is a more complicated structure as a matrix.

For the purpose of this course we consider a matrixto be a second order tensor. This is a defectivedefinition but will serve our purpose.

Higher order tensors can be created by adding more

dimensions to our table as before. Try this:

Matrix and Tensor Functions


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The contraction operation can be performed on

Vectors, matrices and tensors. It is up tp theprogrammer to ensure that these products makesense and that what you are getting out ofMathematica is not simply garbage:

Matrix & Tensor Operations


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Mathematica Generalizes the Inner Product in a

powerful way. The easiest way to understand this isto compare it to the Dot product already seen:

Inner Product


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In the first example, we have the well-known dot

product or contraction. In the second, the actual operation has been slightly

modified. Inner product is far more generalized:

Inner Product


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Sorting a list of scalars is a very simple matter.

Mathematica sorts in ascending order. The Greaterargument can be used to change that default todescending if so desired,:

Sorting


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Study the example on page 98 to get a better grasp

on an application of list sorting.

Sorting Lists


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A 3 by 3 matrix is very useful in Continuum mechanics

that spans the entire scope of Engineering science.This section is for you to get familiar with thecomputation of matrix and tensor functions that arevery useful in this area.

For this, let us recap on some basic definition fromlinear algebra:

Trace The sum of the elements on the diagonal

Matrix Analysis


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There are three invariants of a 3 by 3 matrix. Trace,

Square of the trace minus the trace of the square, andthe determinant.

The characteristics values of a matrix are called theeigenvalues. Check from you math books how togenerate the eigenvalues.

A matrix whose eigenvalues are positive is said to bepositive definite. Many matrices (actually tensors) ofengineering importance are positive definite.

Properties


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Mathematica can be used to compute the

eigenvalues and eigenvectors of your tensor directly.

Properties


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Take the simple example of a 3 by 3 tensor F with a

non-vanishing determinant. The polar decompositiontheorem states that there is a rotation tensor R(Determinant = +1) and positive definite tensors U andV such that

F=RU=VR Given a tensor F, the homework is to find the tensors

R, U and V

Tensor Decompositions


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C=FF and B=FF are the right and left tensors derived

from C. Tensor U is such that UU=C so that U is the square

root of the tensor C.

The Mathematica function MatrixPower can be used

to find U once C is known.

Right and left tensors


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1. Given that =

1

3

2

4

3

0 1 0

0 0 1

Find the rotation tensor, the right stretch tensor and theleft stretch tensor. Demonstrate that the Rotation tensor

is true orthogonal.

Homework


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Find the eigenvalues and eigenvectors of C

Demonstrate numerically that the eigenvalues of Band C are the same.

Demonstrate that the eigenvectors of C and U are thesame

Show that the eigenvectors of B and V are the same.

Show that the determinant of each tensor is simplythe product of the eigenvalues.

HW Continued


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2. Explain the meaning of the program:

HW continued


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Find C, B, R, U, V as in the previous problem in this

case also.


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Spectral Decomposition

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SSG 314 Lists & Other Structures