Post on 05-Apr-2018
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TOPIC: - BIOPOLYMERS.
SUBMITTED TO: SUBMITTED BY:
JAGANJOT KAUR
(SUBJECT TEACHER) GURMUKH SINGH
SECTION: E4001
ROLL NO.: B42
REGD.NO:-11006705
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ACKNOWLEDGEMENT
Thanks giving is a sacred Indian culture. So, first of all, I would liketo thank our subject teacher, MISS JAGANJOT KAUR for his
humble support and encouragement which enhanced me through the
project. His likeness towards my topic uplifted my spirit which in
turn helped me throughout.
Secondly, I express my gratitude to my Parents for being a
continuous source of encouragement and the financial aid given tome.
Thirdly, I would love to thank my beloved friends who
helped me in their little ways. They were of great help and support as
they helped me a lot with their innovative ideas. They helped me a lot
in completing my work in time.
Finally, I would like to thank God for showering his
blessings upon me.
GURMUKH SINGH
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CONTENTS
1. INTRODUCTION 12. HISTORY 23. ALGORITHM OVERVIEW 34. EXAMPLES 3-55. APPLICATIONS 6-96. SYSTEM OF LINEAR EQUATIONS 10-157. GAUSS-JORDAN METHOD 15-168. EXAMPLES 17-189. CONCLUSION 19
REFERENCES
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INTRODUCTION
In linear algebra, Gaussian elimination is an algorithm for solving systems of
linear equations, finding the rank of a matrix, and calculating the inverse of an
invertible square matrix. Gaussian elimination is named after German
mathematician and scientist Carl Friedrich Gauss, which makes it an example of
Stigler's law.
Elementary row operations are used to reduce a matrix to row echelon form.
GaussJordan elimination, an extension of this algorithm, reduces the matrix
further to reduced row echelon form. Gaussian elimination alone is sufficient for
many applications, and is cheaper than the -Jordan version.
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History
The method of Gaussian elimination appears in Chapter Eight, Rectangular
Arrays, of the important Chinese mathematical text Jiuzhang suanshu or The
Nine Chapters on the Mathematical Art. Its use is illustrated in eighteen
problems, with two to five equations. The first reference to the book by this title
is dated to 179 CE, but parts of it were written as early as approximately 150
BCE.[1]
It was commented on by Liu Hui in the 3rd century.
The method in Europe stems from the notes of Isaac Newton.[2]In 1670, he wrote
that all the algebra books known to him lacked a lesson for solving simultaneous
equations, which Newton then supplied. Cambridge University eventually
published the notes as Arithmetica Universalis in 1707 long after Newton left
academic life. The notes were widely imitated, which made (what is now called)Gaussian elimination a standard lesson in algebra textbooks by the end of the
18th century. Carl Friedrich Gauss in 1810 devised a notation for symmetric
elimination that was adopted in the 19th century by professional hand computers
to solve the normal equations of least-squares problems. The algorithm that is
taught in high school was named for Gauss only in the 1950s as a result of
confusion over the history of the subject.
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Algorithm overview
The process of Gaussian elimination has two parts. The first part (Forward
Elimination) reduces a given system to either triangular or echelon form, or
results in a degenerate equation with no solution, indicating the system has nosolution. This is accomplished through the use ofelementary row operations. The
second step uses back substitution to find the solution of the system above.
Stated equivalently for matrices, the first part reduces a matrix to row echelon
form using elementary row operations while the second reduces it to reduced
row echelon form, or row canonical form.
Another point of view, which turns out to be very useful to analyze the
algorithm, is that Gaussian elimination computes matrix decomposition. The
three elementary row operations used in the Gaussian elimination (multiplying
rows, switching rows, and adding multiples of rows to other rows) amount to
multiplying the original matrix with invertible matrices from the left. The first
part of the algorithm computes LU decomposition, while the second part writes
the original matrix as the product of a uniquely determined invertible matrix
and a uniquely determined reduced row-echelon matrix.
Example:
Suppose the goal is to find and describe the solution(s), if any, of the following
system of linear equations:
The algorithm is as follows: eliminate x from all equations below L1, and then
eliminate y from all equations below L2. This will put the system into triangular
form. Then, using back-substitution, each unknown can be solved for.
In the example, x is eliminated from L2 by adding to L2. X is then eliminated
from L3 by adding L1 to L3. Formally:
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The result is:
Now y is eliminated from L3 by adding 4L2 to L3:
The result is:
This result is a system of linear equations in triangular form, and so the first part
of the algorithm is complete.
The last part, back-substitution, consists of solving for the known in reverse
order. It can thus be seen that
Then, z can be substituted into L2, which can then be solved to obtain
Next, z and y can be substituted into L1, which can be solved to obtain
The system is solved.
Some systems cannot be reduced to triangular form, yet still have
at least one valid solution: for example, if y had not occurred in L2 and L3 after
the first step above, the algorithm would have been unable to reduce the system
to triangular form.
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However, it would still have reduced the system to echelon form. In this case, the
system does not have a unique solution, as it contains at least one free variable.
The solution set can then be expressed parametrically (that is, in terms of the
free variables, so that if values for the free variables are chosen, a solution will be
generated).
In practice, one does not usually deal with the systems in terms of equations but
instead makes use of the augmented matrix (which is also suitable for computer
manipulations). For example:
Therefore, the Gaussian Elimination algorithm applied to the augmented matrix
begins with:
Which, at the end of the first part (Gaussian elimination, zeros only under the
leading 1) of the algorithm, looks like this:
That is, it is in row echelon form.
At the end of the algorithm, if the GaussJordan elimination (zeros under and
above the leading 1) is applied:
That is, it is in reduced row echelon form, or row canonical form.
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APPLICATIONS
Finding the inverse of a matrix
Suppose A is a matrix and you need to calculate its inverse. The
identity matrix is augmented to the right of A, forming a matrix
(the block matrix B = [A, I]). Through application of elementary row operations
and the Gaussian elimination algorithm, the left block of B can be reduced to the
identity matrix I, which leaves A 1
in the right block of B.
If the algorithm is unable to reduce A to triangular form, then A is not
invertible.
General algorithm to compute ranks and bases
The Gaussian elimination algorithm can be applied to any matrix A. If
we get "stuck" in a given column, we move to the next column. In this way, for
example, some matrices can be transformed to a matrix that has a
reduced row echelon form like
(The *'s are arbitrary entries). This echelon matrix T contains a wealth of
information about A: the rank of A is 5 since there are 5 non-zero rows in T; the
vector space spanned by the columns of A has a basis consisting of the first,
third, fourth, seventh and ninth column of A (the columns of the ones in T), and
the *'s tell you how the other columns of A can be written as linear combinations
of the basis columns.
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Analysis
Gaussian elimination to solve a system of n equations for n unknowns
requires n (n+1) / 2 divisions, (2n3
+ 3n2 5n)/6 multiplications, and (2n
3+ 3n
2
5n)/6 subtractions, for a total of approximately 2n3 / 3 operations. So it has a
complexity of .
This algorithm can be used on a computer for systems with thousands of
equations and unknowns. However, the cost becomes prohibitive for systems
with millions of equations. These large systems are generally solved using
iterative methods. Specific methods exist for systems whose coefficients follow a
regular pattern (see system of linear equations).
The Gaussian elimination can be performed over any field.
Gaussian elimination is numerically stable for diagonally dominant or positive-
definite matrices. For general matrices, Gaussian elimination is usually
considered to be stable in practice if you use partial pivoting as described below,
even though there are examples for which it is unstable.
Higher order tensors
Gaussian elimination does not generalize in any simple way to higher order
tensors (matrices are order 2 tensors); even computing the rank of a tensor of
order greater than 2 is a difficult problem.
Gaussian elimination is a method for solving matrix equations of
the form
(1)
To perform Gaussian elimination starting with the system of equations
(2)
compose the "augmented matrix equation"
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(3)
Here, the column vector in the variables is carried along for labeling the matrix
rows. Now, perform elementary row operations to put the augmented matrix
into the upper triangular form
(4)
Solve the equation of the th row for , then substitute back into the equation of
the st row to obtain a solution for , etc., according to the formula
(5)
In Mathematical, Row Reduce performs a version of Gaussian elimination, with
the equation being solved by
Gaussian Elimination [m_?MatrixQ, v_?VectorQ] :=
Last /@ Row Reduce [Flatten /@ Transpose [{m, v}]]
LU decomposition of a matrix is frequently used as part of a Gaussian
elimination process for solving a matrix equation.
A matrix that has undergone Gaussian elimination is said to be in echelon form.
For example, consider the matrix equation
(6)
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In augmented form, this becomes
(7)
Switching the first and third rows (without switching the elements in the right-
hand column vector) gives
(8)
Subtracting 9 times the first row from the third row gives
(9)
Subtracting 4 times the first row from the second row gives
(10)
Finally, adding times the second row to the third row gives
(11)
Restoring the transformed matrix equation gives
(12)
Which can be solved immediately to give , back-substituting to obtain
(which actually follows trivially in this example), and then again back-
substituting to find
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Systems of Linear Equations: Gaussian Elimination
It is quite hard to solve non-linear systems of equations, while linear systems are
quite easy to study. There are numerical techniques which help to approximatenonlinear systems with linear ones in the hope that the solutions of the linear
systems are close enough to the solutions of the nonlinear systems. We will not
discuss this here. Instead, we will focus our attention on linear systems.
For the sake of simplicity, we will restrict ourselves to three, at most four,
unknowns. The reader interested in the case of more unknowns may easily
extend the following ideas.
Definition. The equation
a x + b y + c z + d w = h
Where a, b, c, d, and h are known numbers, while x, y, z, and w are unknown
numbers, is called a linear equation. If h =0, the linear equation is said to be
homogeneous. A linear system is a set of linear equations and a homogeneous
linear system is a set of homogeneous linear equations.
For example,
And
Are linear systems, while
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is a nonlinear system (because of y2
). The system
Is a homogeneous linear system.
Matrix Representation of a Linear System
Matrices are helpful in rewriting a linear system in a very simple form. The
algebraic properties of matrices may then be used to solve systems. First,
consider the linear system
Set the matrices
Using matrix multiplications, we can rewrite the linear system above as the
matrix equation
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As you can see this is far nicer than the equations. But sometimes it is worth to
solve the system directly without going through the matrix form. The matrix A is
called the matrix coefficient of the linear system. The matrix C is called the
nonhomogeneous term. When , the linear system is homogeneous. The
matrix X is the unknown matrix. Its entries are the unknowns of the linear
system. The augmented matrix associated with the system is the matrix [A|C],
where
In general if the linear system has n equations with m unknowns, then the matrix
coefficient will be a nxm matrix and the augmented matrix an nx (m+1) matrix.
Now we turn our attention to the solutions of a system.
Definition. Two linear systems with n unknowns are said to be equivalent if and
only if they have the same set of solutions.
This definition is important since the idea behind solving a system is to find an
equivalent system which is easy to solve. You may wonder how we will come up
with such system. Easy, we do that through elementary operations. Indeed, it is
clear that if we interchange two equations, the new system is still equivalent to
the old one. If we multiply an equation with a nonzero number, we obtain a new
system still equivalent to old one. And finally replacing one equation with the
sum of two equations, we again obtain an equivalent system. These operationsare called elementary operations on systems. Let us see how it works in a
particular case.
Example. Consider the linear system
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The idea is to keep the first equation and work on the last two. In doing that, we
will try to kill one of the unknowns and solve for the other two. For example, if
we keep the first and second equation, and subtract the first one from the last
one, we get the equivalent system
Next we keep the first and the last equation, and we subtract the first from the
second. We get the equivalent system
Now we focus on the second and the third equation. We repeat the same
procedure. Try to kill one of the two unknowns (y or z). Indeed, we keep the first
and second equation, and we add the second to the third after multiplying it by 3.We get
This obviously implies z = -2. From the second equation, we get y = -2, andfinally from the first equation we get x = 4. Therefore the linear system has one
solution
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Going from the last equation to the first while solving for the unknowns is called
backsolving.
Keep in mind that linear systems for which the matrix coefficient is upper-triangular are easy to solve. This is particularly true, if the matrix is in echelon
form. So the trick is to perform elementary operations to transform the initiallinear system into another one for which the coefficient matrix is in echelon
form.
Using our knowledge about matrices, is there any way we can rewrite what we
did above in matrix form which will make our notation (or representation)
easier? Indeed, consider the augmented matrix
Let us perform some elementary row operations on this matrix. Indeed, if we
keep the first and second row, and subtract the first one from the last one we get
Next we keep the first and the last rows, and we subtract the first from the
second. We get
Then we keep the first and second row, and we add the second to the third after
multiplying it by 3 to get
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This is a triangular matrix which is not in echelon form. The linear system for
which this matrix is an augmented one is
As you can see we obtained the same system as before. In fact, we followed the
same elementary operations performed above. In every step the new matrix was
exactly the augmented matrix associated to the new system. This shows that
instead of writing the systems over and over again, it is easy to play around with
the elementary row operations and once we obtain a triangular matrix, write the
associated linear system and then solve it. This is known as Gaussian
Elimination.
GAUSS-JORDAN METHOD:
The Gauss-Jordan method is a version of Gaussian elimination in solving
systems of linear equations. The variables' coefficients, instead of merely being
reduced to a triangular shape, are reduced to a diagonal. This eliminates the
need for successive substitution, allowing one to just read off the solutions.
Gauss-Jordan Elimination
Gauss-Jordan elimination goes the extra step of using such operations to
eliminate variables above the diagonal as well.
As a result, one can just read off the solution, for example, that x1 = -1, x2
= 2, and so on. The need for back-substitution to solve for each variable,
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As in Gaussian substitution, is therefore eliminated.
Difference from Gaussian Elimination
The additional operations Gauss-Jordan performs to put the
variables into a diagonal form triples the number of computations
required, even with Gaussian elimination's back-substitution operations.
The gain, however, is in being able to read answers off immediately.
Disadvantages
1. The additional operations of Gauss-Jordan add to rounding error andcomputer time. A disadvantage of both Gaussian and Gauss-Jordan
elimination is that they require the right vector, for example, (4,1,-3,4)
above, to be known. If these numbers are to be learned later, a method
called matrix factorization can prepare a triangular shape for easy
calculation when the vector is known. If the vector changes, the effort in
factorizing has saved time as well.
Elimination Method for Solving Systems of LinearEquations
A technique for solving systems of linear equations of any size is the Gauss-
Jordan Elimination Method. This method uses a sequence of operations on a
system of linear equations to obtain an equivalent system at each stage. An
equivalent system is a system having the same solution as the original system.
The operations of the Gauss-Jordan method are:1. Interchange any two equations.
2. Replace an equation by a nonzero constant multiple of itself.
3. Replace an equation by the sum of that equation and a constant multiple of
any other equation.
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EXAMPLES:
1. Let's solve the system:-2x
1- 3x
2= -6
5x1 + 4x2 = 31
We form the matrix (A|b) and reduce it:
This is the matrix (A|b).
Row 1:= 1/2 x row 1
Row 2:= row 2 - 5 x row1
Row 2:= 2/23 x row 2
Row 1:= row 1 + 3/2 row 2
The reduced matrix represents the system
1x1 + 0x2 = 3
0x1 + 1x2 = 4
Which has the unique solution (x1,x2) = (3,4). This is also the unique
solution to the original system.
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2. Let's solve the system
x1 + x2 + x3 = 5
x1 + 2x2 + 3x3 = 8
We form the matrix (A|b) and reduce it:
This is the matrix (A|b)
Row 2:= row2 - row1 the matrix is in echelon form
row1:= row1- row2 the matrix is reduced
The reduced matrix represents the system
1x1 + 0x2 - x3 = 2
0x1 + 1x2 + 2x3 = 3
Which has infinitely many solutions:
x1 = 2 + x3, x2 = 3 - 2x3, x3 = any number
We wrote the solution to the exercise on the before page as two equations:
X1 = 2 + x3, x2 = 3 - 2x3, (x3 R)
Notice that these two equations have the perfect same meaning as one
vector equation:
When we write the equation this way, it looks clear that the solution set is
the line through the point (2,3,0) that is parallel to the vector (1, -2, 1).
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CONCLUSION
Gaussian elimination is a method of solving a linear system (consisting
of equations in unknowns) by bringing the augmented matrix. In the
Gaussian Elimination Method, Elementary Row Operations (E.R.O.'s) are
applied in a specific order to transform an augmented matrix into triangular
echelon form as efficiently as possible.
This is the essence of the method: Given a system of m equations in n variables
or unknowns, pick the first equation and subtract suitable multiples of it from
the remaining m-1 equations. In each case choose the multiple so that the
subtraction cancels or eliminates the same variable, say x1. The result is that the
remaining m-1 equations contain only n-1 unknowns (x1 no longer appears).
Now set aside the first equation and repeat the above process with the remaining
m-1 equations in n-1 unknowns.
Continue repeating the process. Each cycle reduces the number of variables and
the number of equations. The process stops when either:
There remains one equation in one variable. In that case, there is a uniquesolution and back-substitution is used to find the values of the other
variables.
There remain variables but no equations. In that case there is no uniquesolution.
There remain equations but no variables (i.e. the lowest row(s) of theaugmented matrix contain only zeros on the left side of the vertical line).
This indicates that either the system of equations is inconsistent or
redundant. In the case of inconsistency the information contained in the
equations is contradictory. In the case of redundancy, there may still be a
unique solution and back-substitution can be used to find the values of the
other variables.
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REFERENCES
1. HIGHER ENGINEERING MATHEMATICSBY B.V. RAMANA.
2. MODERN APPROACH TO MATHEMATICSBY N.K. NAG.
3. S. CHAND MATHEMATICS FOR CLASS XIIBY S. CHAND PUBLICATIONS.
4. www.google.com
http://www.google.com/http://www.google.com/