수치해석 (numerical analysis) 행렬과 연립 방정식 (part 2) 문양세 강원대학교 ...

Discrete Mathematics

(Numerical Analysis)

(Part 2)

IT Page 2We are now LU Decomposition & Simultaneous EquationNumerical Analysisby Yang-Sae Moon2Page 3 LU Decomposition & Simultaneous Equation , . (: ) .Ax = b A b , - . 1) ,2) b x . Numerical Analysisby Yang-Sae Moon3Page 4 ?LU Decomposition & Simultaneous Equation A L U .

,L (lower triangular matrix),U (upper triangular matrix).

L U ( ) .Numerical Analysisby Yang-Sae Moon4Page 5 L U (1/5)LU Decomposition & Simultaneous Equation L , L A .

Numerical Analysisby Yang-Sae Moon5Page 6 L U (2/5)LU Decomposition & Simultaneous Equation U

Numerical Analysisby Yang-Sae Moon6Page 7 L U (3/5)LU Decomposition & Simultaneous Equation L

Numerical Analysisby Yang-Sae Moon7Page 8 L U (4/5)LU Decomposition & Simultaneous Equation U

Numerical Analysisby Yang-Sae Moon8Page 9 L U (5/5)LU Decomposition & Simultaneous Equation , .

Numerical Analysisby Yang-Sae Moon9Page 10 (1/4)LU Decomposition & Simultaneous Equation A L U .

, li1 u1j .

Numerical Analysisby Yang-Sae Moon10Page 11 (2/4)LU Decomposition & Simultaneous Equation, li2 u2j .

Numerical Analysisby Yang-Sae Moon11Page 12 (3/4)LU Decomposition & Simultaneous Equation, li3 u3j .

, li4 .

Numerical Analysisby Yang-Sae Moon12Page 13 (4/4)LU Decomposition & Simultaneous Equation, A L U .

, L U LU A .

Numerical Analysisby Yang-Sae Moon13Page 14 LU Decomposition & Simultaneous Equation A, L, U . ( 9 )

L ( 8 )., A .

Observation: A , L 0 .Numerical Analysisby Yang-Sae Moon14Page 15 - LU Decomposition & Simultaneous Equationprocedure LUmatrices(aij: real numbers, n: integer){ [aij] is an nxn matrix. (1 i,j n)}{ n is # of columns(= # of rows).}Initialize every lij in [lij] and every uij in [uij] to 0;for m := 1 to nj := m;for i := j to nlij := end

i := m;for j := i to nuij := endend

return [lij] and [uij];

Numerical Analysisby Yang-Sae Moon15Page 16LU Decomposition & Simultaneous Equation (1/3)



Numerical Analysisby Yang-Sae Moon18Page 19LU Decomposition & Simultaneous Equation I (1/2)

Numerical Analysisby Yang-Sae Moon19Page 20LU Decomposition & Simultaneous Equation I (2/2) ( p. 153 )

Numerical Analysisby Yang-Sae Moon20Page 21LU Decomposition & Simultaneous Equation II (1/2)

Numerical Analysisby Yang-Sae Moon21Page 22LU Decomposition & Simultaneous Equation II (2/2) ( p. 161 )

Numerical Analysisby Yang-Sae Moon22Page 23We are now LU Decomposition & Simultaneous EquationNumerical Analysisby Yang-Sae Moon23Page 24 (1/6)LU Decomposition & Simultaneous Equation ? , Numerical Analysisby Yang-Sae Moon24Page 25 (2/6)LU Decomposition & Simultaneous Equation() .

A L U .

.

Numerical Analysisby Yang-Sae Moon25Page 26 (3/6)LU Decomposition & Simultaneous Equation . (forward substitution) .

Numerical Analysisby Yang-Sae Moon26Page 27 (4/6)LU Decomposition & Simultaneous Equation, y L b .

y , y U , x .Numerical Analysisby Yang-Sae Moon27Page 28 (5/6)LU Decomposition & Simultaneous Equation . (backward substitution) .

Numerical Analysisby Yang-Sae Moon28Page 29 (6/6)LU Decomposition & Simultaneous Equation, x U y .

Numerical Analysisby Yang-Sae Moon29Page 30LU Decomposition & Simultaneous Equation . (1/2)

.

Numerical Analysisby Yang-Sae Moon30Page 31LU Decomposition & Simultaneous Equation( ) L y . (2/2)( ) U x .

Numerical Analysisby Yang-Sae Moon31Page 32LU Decomposition & Simultaneous Equation procedure LUequation(aij, bi: real numbers, n: integer){ [aij] is an nxn matrix for coefficients. (1 i,j n)}{ [bi] is an nx1 matrix for results. (1 i n)}{ n is # of columns(= # of rows).}[lij], [uij] := LUmatrices(aij, n); // get matrices L and U

for i := 1 to nyi :=

for i := n to 1xi :=

return [xi];






Numerical Analysisby Yang-Sae Moon37Page 38LU Decomposition & Simultaneous Equation I (1/2)

Numerical Analysisby Yang-Sae Moon38Page 39

LU Decomposition & Simultaneous Equation I (2/2)

Numerical Analysisby Yang-Sae Moon39Page 40LU Decomposition & Simultaneous Equation II (1/2)


LU Decomposition & Simultaneous Equation II (2/2)

Numerical Analysisby Yang-Sae Moon41Page 42We are now LU Decomposition & Simultaneous EquationNumerical Analysisby Yang-Sae Moon42Page 43 (1/6)LU Decomposition & Simultaneous Equation . . . .

( )(upper triangular) ,(lower triangular) .Numerical Analysisby Yang-Sae Moon43Page 44 (2/6)LU Decomposition & Simultaneous Equation L , Z .

Numerical Analysisby Yang-Sae Moon44Page 45 (3/6)LU Decomposition & Simultaneous Equation L Z .

Z () .

Numerical Analysisby Yang-Sae Moon45Page 46 (4/6)LU Decomposition & Simultaneous Equation L Z .

Z () .

Numerical Analysisby Yang-Sae Moon46Page 47 (5/6)LU Decomposition & Simultaneous Equation , Z .

Numerical Analysisby Yang-Sae Moon47Page 48 (6/6)LU Decomposition & Simultaneous Equation, . , V .

Z , V .

p. 168 vij (u v, index )Numerical Analysisby Yang-Sae Moon48Page 49 (1/2)LU Decomposition & Simultaneous Equation A .

A L U .

Numerical Analysisby Yang-Sae Moon49Page 50 (2/2)LU Decomposition & Simultaneous Equation , L U . L-1 U-1 , A .

Numerical Analysisby Yang-Sae Moon50Page 51 LU Decomposition & Simultaneous Equationprocedure LUinverse(aij: real numbers, n: integer){ [aij] is an nxn matrix for coefficients. (1 i,j n)}{ n is # of columns(= # of rows).}

[lij], [uij] := LUmatrices(aij, n); // get matrices L and U

Set zij and vij to 0 for every i and j;

for j := 1 to n zjj := 1/ljj; for i := j+1 to nzij :=

for i := 1 to nvii := 1for j := i+1 to n vij :=

return [vij][zij];

Numerical Analysisby Yang-Sae Moon51Page 52 (1/5)LU Decomposition & Simultaneous Equation





Numerical Analysisby Yang-Sae Moon56Page 57 I (1/2)LU Decomposition & Simultaneous Equation ( p. 174 )

Numerical Analysisby Yang-Sae Moon57Page 58 I (2/2)LU Decomposition & Simultaneous Equation

Numerical Analysisby Yang-Sae Moon58Page 59 II (1/2)LU Decomposition & Simultaneous Equation


II (2/2)LU Decomposition & Simultaneous Equation

Numerical Analysisby Yang-Sae Moon60Page 61Homework #3LU Decomposition & Simultaneous EquationNumerical Analysisby Yang-Sae Moon61

수치해석 (numerical analysis) 행렬과 연립 방정식 (part 2) 문양세 강원대학교 ...

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