linear system of solutions

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PRESENTED BY

(14MD02) ANANTHA BABU (ME ED)(14MI31) ASWATH S (ME CIM)

(14MI01) AMAL MADHU M (ME CIM)

AMMS SEMINAR

SOLUTION OF LINEAR SYSTEMS

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NEW DECOMPOSITION METHOD FOR SOLVING ASYMMETRIC MATRIX

A RESEARCH PAPER BYJENN-CHING Luo Department of Civil Engineering and Engineering Mechanics, Columbia University, New York

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SOLVING SYSTEM OF EQUATIONS

LINEAR NON LINEAR

NON ITERATIVE (or) NEWTON RAPHSON DIRECT METHOD METHOD

SOLVING DECOMPOSITION VARIABLE

GAUSS CHOLESKIELIMINATION METHOD

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Solving a system of equations is divided into:-

1. Linear equations:- ax=b

A common form of a linear equation in the two variables x and y is

y=mx+b

2. Non linear equations:- Equation whose graph does not form a straight

4x2 + 2y - 1 = 0 and x3 + 2x2 - 4xy - 1 = 0 are the examples of nonlinear equations.

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• Gauss Elimination Method

Advantages:-

Easy computation.

Less time consuming

Less memory usage

Disadvantages:-

Division by zero is default error

The matrix should be a non-singular matrix.

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• Decomposition (Choleski method):-

direct method.

for symmetric matrix.

Advantages:-

Computations are easy.

Disadvantages:-

This method can be used only for symmetric matrix.

We may get imaginary values which will be difficult to solve.

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Relaxation method:-It is used for solving system of linear equations.

It is an iterative method.

The necessary condition to solve this matrix is that the matrix should be diagonally dominant which should be numerically largest.

Advantages:-It is a sparse matrix.

Disadvantages:-

It is very slow process.

Truncation and propagation error may occur.

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JENN-CHING Luo’s METHOD

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• An innovative decomposition for inverting a nonsingular, asymmetric, matrix [A] of

order (n X n)

• This technique requires a minimal amount of computer memories

• Tries to overcome the limitations in Cholesky method and tedious calculations in relaxation method

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• [A]¯¹ = [L] [D] [U]

[L] = lower triangular matrix, [D] = diagonal matrix, and [U] = upper triangular matrix

• [X] = [A]¯¹ [B] becomes [X] = [L] [D] [U] [B]

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• Consider a lower triangular matrix [L] of order (n × n) with unit diagonal coefficients

[L] = [(L1) (L2).. (Lj)... (Ln)]

Where (Lj) is jth column vector of [L]

• Lij the ith coefficient of {Lj}, in which Lij = 0 for i < j and Ljj = 1

METHODOLOGY

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• Upper triangular matrix [U] of order (n x n) with unit diagonal coefficients

U= [ {U1} {U2} {U3}

. .

{Uj} . .

{Un} ]

• Uij = 0 for i > j and Ujj = 1.

{Uj} is the jth row vector of [U]

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• Consider a system of linear equations as

[A][X] = [B]…………………………………………………………..(1)A is Non-singular square matrix which may be asymmetric.• [X] = [L][X’]…………………..………………………………………(2)

transformation in term of [L]

• [A][L][X’] = [B].From (1) & (2)

Multiplying LHS and RHS by [U]

• [U] [A] [L] [X’] = [U][B]………………………………………..(3)

• [U] [A] [L] = [D-1]…………………………………………………(4)

If [U] [A] [L] is a diagonal matrix, then [U] [A] [L] may be easily inverted

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• [X’] = [D][U][B]OR

• [X] = [L] [D] [U] [B]

( As [X]=[A-1] [B]) Then,

• [A]-1=[L] [D] [U]………………………………………………………(5)

• If we can find U and L such that U*A*L is a diagonal matrix, then inverse can be found.

• [Ui][A](Lj) = 0 , i!=j ; [Ui][A](Lj)!=0 , i=j ……………………………………condition

i.e off-diagonal coefficients of [U] [A] [L] should be 0.

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• [U][A][L] should be diagonal and invertible, For [A]¯¹ = [L] [D] [U]

• PROOF 1 (diagonal character)

Upper and lower Triangular matrix elements written as

{Uj} = (ej) +∑ri [Ui]…………………………….. i j+1 to n……………………………….(6)

(Li) = (fj) + ∑si [Li] …………………………….. i j+1 to n……………………………….(7)

(ej), a unit row vector with one non-zero coefficient in the jth entry, (fj), an unit column vector with one non-zero coefficient in the jth entry,

ri and si are coefficients to be determined.

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• Kth coefficient of {Uj}

Ujk = ejk + ∑ ri Uik………………………………………. ij+1 to n

when j > k ; ejk = 0 and lower bound of i , is (j + 1), which means i >_ j + 1 > k + 1 > k such that Uik = 0

• This means Ujk = 0, when j > k………………………………(8)

when k = j ; Ujj = 1 + ∑ ri Uij

Since i > j + 1 , Uij = 0

Ujj = 1………………………………….(9)

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From (8) and (9)

the requirement for the upper triangular configuration is satisfied.

lll ly for (L) vector, it can be proved.

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• The unknowns in Equation (6) and (7) Slide 14 are ri and si i.e 2(n-j) unknowns

• Using conditions Slide 13

{Ui}[A](Lj) = 0 i j+1 to n……………………………………………….(10)

{Uj}[A](Li) = 0 i j+1 to n……………………………………………….(11)

• Equations in ri and si are obtained.

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Consider (6) and (7) equations

{Uj} = (ej) +∑ri [Ui]…………………………….. i j+1 to n……………………………….(6)

(Li) = (fj) + ∑si [Li] …………………………….. i j+1 to n……………………………….(7)

In (6), multiply with [A](Lk) in LHS and RHS, we get

{Uj}[A](Lk) = [Aj] (Lk) + ∑ri [Ui] [A](Lk)

Using conditions ; LHS = 0 , and RHS simplifies to ri [Ui] [A](Li) where (k=i)…………………………………....(13)

where (k > j)……………………………………(12)

{Uj} = (ej) +∑ri [Ui]…………………………….. i j+1 to n……………………………….(6)

(Li) = (fj) + ∑si [Li] …………………………….. i j+1 to n……………………………….(7)

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• Eqn (12) simplified as

[Aj] (Lk) + ri [Ui] [A](Li) = 0

ri = - [Aj] (Lk) / [Ui] [A](Li)…………………………………………………………..(14)

or

• lll ly Si = - {Ui} [Aj] / [Ui] [A](Li)…………………………………………………………..(15)

From (10),(11) and (14),(15) we can solve for ri and si and can get [U} and [L] for the corresponding [A] matrix

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{Uj} = (ej) +∑ri [Ui]…………………………….. i j+1 to n……………………………….(6)

(Li) = (fj) + ∑si [Li] …………………………….. i j+1 to n……………………………….(7)

Substituting ri and si in (6) and (7), we get values of

each row vector of upper triangular matrix and

each column vector of lower triangular matrix

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Hence

• [U] and [L] is determined.

• U*A*L is proved to be diagonal matrix

• [A]-1=[L] [D] [U]

• [X] = [L] [D] [U] [B]

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SUMMING UP - CONCLUSION

PROS

Factorization of asymmetric matrix possible

Less Computational memory than Relaxation and other iterative methods.

Direct method; without the problem of convergence

Can be used where Gauss Siedel method, conjugate method etc fails.

Efficient inversion of asymmetric and indefinite matrices possible

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Naivety present as in Gauss Elimination, so pivoting is necessary.

To limited problems only.

If [A] is a sparse matrix, then general algorithm will bring in fluke arithmetic operations

CONS

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o J.-C. Luo, A new class of decomposition for symmetric systems, Mechanics Research Communications 19

(3) (1992).

o J.-C. Luo, An incomplete inverse as a pre-conditioner for the conjugate gradient method, Computers

Mathematics with Applications (to appear).

o J.-C. Luo, A note on parallel processing, Applied Mathematics Lettera 5 (2), 75-78 (1992).

FUTURE REFERENCE

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THANK YOU