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수치해석 2010 년도 봄학기 Homework PowerPoint PPT Presentation


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수치해석 2010 년도 봄학기 Homework. 환경공학과 20080029 김민진. I.1 Taylor 급수를 유도하고 절단 오차를 설명하라. 절단오차 (truncation error) - 수학적으로 엄밀하게 주어지는 함수 f 의 값을 , 유한의 사칙 연산의 반복 계산 식 fa 로 근사하는 경우 , (fa-f) 의 오차가 생김 . - 예를 들면 , 테일러급수를 이용하여 삼각함수를 계산할 때 , 무한급수의 계산을 유한 항까지의 계산으로 중단하기 위해서 , 절단오차가 발생.

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수치해석 2010 년도 봄학기 Homework

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2010 Homework

20080029


I.1 Taylor .


  • (truncation error)

    - f , fa , (fa-f) .

    - , , , .


I.2 1) .2) Visual Fortran, Visual C Compiler . 3) Visual Basic .


(1)

  • :: .

  • : 0 .

  • : 1 .

  • : .

  • : . .

  • : .


.


Visual Fortran


Visual C


III.1 3 (Determinant) .( ).


  • A LU A=LU , |A| |A|=|L| |U|

  • . Doolittle LU , L 1, ()

  • . , |A| U . ,LU Pivot , Pivot , m .


  • , |A| U . ,LU Pivot , Pivot , m .

  • Doolittle LU , U Gauss (4.14) . , Gauss Pivot


  • . , Gauss , , Pivot . , .


  • A , .

  • ,I . , (4.4) , . ,LU .


  • X , (5.5) AX=I

  • (5.6) . Doolittle LU A=LU LY=I , UX=Y X . , LY=I 1 , , j 1 n


  • . , , j n 1

    . , Y L , X U , . , LU A , X A . , LU Pivot , X . Pivot .


III.2 3 Gauss .


III.3 3 Gauss Pivot .


III.4 3 C Fortran .


III.5 3 Gauss-Jordan .


  • ()

  • 1, 1 1 0 . , 1 1/2 .


  • , 2-1, 3-41

  • 2 , 2 1 0 . , 2 -1/3 .


  • , 1-22, 3-(-7)2

  • 3 , 3 1 0 . , 3 -3/56 .


  • , 1-(13/3)3,

  • 2-(-2/3)3

  • , .


III.6 4 LU Doolittle Crout .


III.7 3 LU Excel .


III.8 1 Jacobi, Gauss Seidel , SOR , Excel, .


IV.1 Derive the all the algorithms using graphs for the case of Bisection, Secant, and Newton Methods.


IV.2 Compile and run the given programs by using Visual Fortran, Visual Basic and Visual C Compilers.


IV.3 Solve the example problems by hands, using Excel program, and by the given programs.


IV.4 Newton Newton-Raphson .


6.2 Newton-Raphson

(1)

(2)

(3)

6.5: 1

(6.5)

(6.6)

Newton-Raphs


  • 6.2.2 Newton-Raphson .

    - Newton-Raphson .

    - (6.5)Newton-Raphson .Newton-Raphson . .

    - (Sol.)

    - 1


- Newton-Raphson

- Newton-Raphson (X)

- = & .

- = .

.


IV.5 Newton . , .


VII.1 BOD ARMA 1 ( ). .


1)

BOD , BOD . .

, (u) (Q/A).

.

.

x=0 Co x C . .

, .

, Co X=0 .


2)

  • .

    BOD / ln y / x . . , BOD . .

    . Excel Regression() .(r)


VIII.1 , , Sympson .


,

:

. , .

X=a, X=b, y=f(x) . , [a,b] n , .

, , . , n , 8.1


VIII. 2 , Sympson (, Excel, Program ( ), . , .


VII.2 ARMA .


VIV.2 Euler 2 Taylor .


VIV.3 Adams-Basforth Adams-Moulton .


I.1 .


I.2 Fortran Basic Excel Excel . . .


2.4 M1.FOR ( )


  • 2.5 M1.FOR ( )


2.5 M1.FOR ( )


2.6 M1.FOR ( )


2.7 M1-Excel


2.8 M1-Excel


I.3 , .


  • (hyperbolic), (parabolic), (elliptic) .

  • .


  • , . . .


  • . . .


  • , .

  • , .

  • GCA , . . , .


  • . . Excel . BASIC FORTRAN . . Crank-Nicholson (GCN : Generalized Crank-Nicholson Method) (GCA : Generalized Characteristic Averaging Method) .


I.4 GCA , Fortran Basic Excel Excel . . .


  • GCA (Characteristic Method), (Centered Difference (Averaging) Method), (Generalized) Crank Nicholson .

  • 1)

    • .


  • , .

  • , .

  • GCA , . . , .


2) () (Centered Difference (Averaging) Method)

  • . , , .


  • .

  • , , .

  • , ().

  • , .


  • . , , . L(C) .


  • 0 . . .


  • (Weighted Residual Method) , . , , , 0 . Green , , .

  • , ( ) . , (, ) , , , . , (Basis Function) (Shape Function) .


  • , . , , 1 . , , , . Gauss , .


II.

II.1 Green .


  • . .


II.2 . .


II.3 .


II.4 Gauss . ( ).


  • Gauss , 2 . Gauss .


  • . . . , .


II.5 .


II.6 , Fortran (m1.for) Excel Excel . . .


  • . . .

  • .


II.7 (gcn.bas, gca.bas, gcn.xls, gca.xls) (m1.for, m1.xls) ( ).


  • Crank-Nicholson . GCN . , .


.


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