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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 .

- :: .
- : 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.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 . , .

.