MATHCAD

2002

-

.. , ..

MATHCAD

2002 004.4 32.973-018

..

..

762 .. MathCAD: . : .. , .. . : , 2002. 136 .

ISBN-5-7795-0169-6

- MathCAD2001i ( ) - (, ) . - MathCAD. . , .

, -, , - .

-

: .. , ..-.., ,

;

.. , ..., ()

ISBN-5-7795-0169-6 , 2002 ..,

.., 2002

MATHCAD

..

- 54. . 02.953. .127.10.01 01.10.2001 .

10.12.2002. 6084 1/16 . . . . .

8 ,0 . - . . ; 8 ,75 . . 400 .

-

630008, , . , 113

3 4

.............................................................................................. 5 1.

MathCAD .............................................................................. 7 1. .................... 7

1.1. MathCAD ....................................... 7 1.2. .......................................................... 9 1.3. ................................................... 13

2. - ......................... 15

2.1. Word ... 15 2.2. Excel .. 19

2. MathCAD ........... 27 3.

MATHCAD ........................................................................ 28 3.1. ...................... 28 3.2. ........ 29 3.3. ................. 35

4. -: ........................................................................... 39

4.1. - ..................................................... 39

4.2. - MathCAD ........... 42 5.

- MathCAD .................... 44 5.1.

- .................................................... 44 5.2.

- ................................................. 45 5.3.

- ................................................. 51 6.

- MathCAD .......... 62 6.1. ........ 62 6.2.

............................................ 66

6.3. ...................... 74 7. MathCAD ..... 80

7.1. ............. 80 7.2.

MathCAD ......................................................... 81 7.3.

MathCAD ................................ 84 7.4. MathCAD Internet ....................................... 85

3. -

MathCAD ....................................................... 94 8.

MathCAD ........................................................... 94 8.1. ...................................... 94 8.2. ............................................. 105

9. MathCAD .......................................................................... 112

9.1. ..... 112 9.2. .... 114

10. MathCAD ..................................................... 118

10.1. ........................................................................... 118

10.2. .................... 125 .................................................................................... 135 ................................................ 135

5 6

, (.. ) ( ) . -, , , , - . . - : (

); ( -

- ); . ( -

Visual) - ( ), - ( ). , 90- , - - [1] , , . MathCAD [2, 3], MatLab [4, 5], Mathematica [6], Maple [7].

, - MathCAD, MathCAD2000 Professional, Math-CAD2001i Professional. - - - , , ( ) Internet.

MathCAD - [2, 8, 9]. , ,

- MathCAD. - , , , MathCAD, . - , - MathCAD.

- . - MathCAD.

2 - MathCAD ( MathCAD2001i), -: , . -- MathCAD. - . - , , - MathCAD.

, MathCAD ( , , , - , -). - , - - .

, , , , - . , -, , MathCAD .

7 8

1. MathCAD

MathCAD, -

MathCAD -, Word Excel.

1.

MathCAD, -

, ASCII-, - 0 255. MathCAD , - (, , Pas-cal ..). MathCAD, , - MathCAD.

1.1. MathCAD , , -

.

, : ; . -

, . . ( ) . - ( ), , .

-:

, , - ;

( [Enter]) ( );

- ( Excel);

, .. - ;

, ASCII-, - ;

, - .

:

, , , - ;

.

, :

, ( );

, -, .

1.1.1. READ :

) : =READ(E :\COPY\data1.dat) ) file_1 : =E :\COPY\data1.dat A : = READ(file_1)

9 10

1.2.

: WRITE ( ) : = APPEND ( ) : = : WRITE ,

. -, ;

APPEND . , , ;

APPEND , ( );

: ; ; ;

- , - (. 1.2.1).

1.2.1. : 10=sampn ; sampn [0, 99]; sampn 0 1sampn . ,

, . 1.2.1. rnd(a) , [0, a], floor(x) , x. .

nsamp 10:= i 0 nsamp 1..:=xi floor rnd 100( )( ):=WRITE file_1( ) nsamp:=APPEND file_1( ) xi:=APPEND file_1( ) i:=

. 1.2.1. 1.2.1. -

MathCAD ( MathCAD2000 ). - MathCAD2001i - (. . 1.2.2), .

. 1.2.2. WRITE

: WRITEPRN ( ) := APPENDPRN ( ) :=

:

WRITEPRN -, ;

APPENDRN . - ,

11 12

. - ;

, ;

: ; ; .

, , .

- :

PRNCOLWIDTH , ( 8);

PRNPRECISION ( 4).

, , , .

1.2.2. , k- - 100 = 20 = ( 420 = ).

, - , . 1.2.3. - ( n = 5, m = 4).

0 10:= 0 2:=n 5:= m 4:=k 0 m 1..:=B k rnorm n 0, 0,( ):=WRITEPRN file_2( ) B:=

. 1.2.3.

1.2.1. MathCAD, -

, : nixi ,0, = -

:

n

abaxi+= ;

nixi ,0),( = , )5(cos)(

2xex x = .

1.2.2. MathCAD, - , , - 1.2.2, m8 , k- - 100 = - 1,0,01.02 =+= mkkk .

1.2.2. MathCAD -

, ASCII-. , Windows (- ), - PASCAL, .

1.2.3. , -

1.2.2.

13 14

1.3.

- :

:= READ(), , (.. ).

, READ . - , (. 1.3.1).

1.3.1. z m - _1.data dat , 1.2.1. m _1.data dat . . 1.3.1.

file_1 "F:\COPY\data_1.dat":=m READ file_1( ):= m 10=

i 0 2 m..:= rabi READ file_1( ):= n last rab( ):=j 0 m 1..:= z j rabn m j+ 1+:=

zT 0 1 2 3 4 5 6 7 8 90 0 1 2 3 4 5 6 7 8 9

=

. 1.3.1. 1.3.1. -

MathCAD ( MathCAD2000 ). - MathCAD2001i - .

:

:= READPRN(),

, . : ; , ASCII-, -

.

1.3.2. - , 1.2.2.

. 1.3.2. mean , var . - . .

B READPRN file_2( ):=n rows B( ):= m cols B( ):= n 5= m 4=j 0 n 1..:= k 0 m 1..:=x j k n+ B j k,:= mean x( ) 10.174= var x( ) 3.215=

. 1.3.2.

1.3.1. , V, k- k- , 1.2.2 file_2.dat.

, MathCAD -.

15 16

Simpson f a, b, N,( ) h b aN

S f a( )( ) f b( )+

S S 4 f a i h+ h2

++

i 0 N 1..for

S S 2 f a i h+( )+i 1 N 1..for

h6

S

:=

2. -

-

MathCAD, - MathCAD (-) Word ( ) Ex-cel.

2.1. Word , ,

- , , - MathCAD. .

1. MathCAD Word.

2. Word MathCAD.

, - - Word - -. .

: 1. MathCAD

, Word.

: ) MathCAD,

, Copy ();

) Word , ;

) MathCAD, Paste () .

, -, .. Word. Word , - . : ) , ( ), , ).

, - . , , . .

. 2.1.1 Word, MathCAD - - . - MathCAD Word.

- - ( ).

-:

f - ;

a, b -;

N - -

. 2.1.1. MathCAD

2. Word -

MathCAD,

17 18

. - MathCAD, :

) , ; ) (. 2.1.2) -

MathCAD document ;

. 2.1.2. ) ( ) -

(-, , - . 2.1.3);

) MathCAD - MathCAD. Word MathCAD (. . 2.1.4);

) MathCAD - MathCAD .

. 2.1.3. MathCAD Word

0

1xx2 cos 4 x( ) d 0.247=

. 2.1.4. MathCAD Word

MathCAD . , ) - (. . 2.1.2) . , - , - MathCAD, Word , - . 2.1.5.

. 2.1.5. MathCAD -

Word MathCAD , ( - ) Math-CAD.

Mathcad Document

19 20

MathCAD MathCAD , - Word.

Word - , MathCAD. MathCAD -, ) ) 1.

2, MathCAD - Word. :

Word (. . 2.1.6, , Word);

Word ( 2).

2.1.1. Math-CAD - ( -, ). - (. . - OLE).

2.2. Excel , Math-

CAD Excel , - ( . 2.1 Word).

, - MathCAD ( MathCAD2000), AddIn for Excel, MathCAD Excel. - MathCAD. , , -, MathCAD.

. 2.1.5. MathCAD Word 1. Excel ,

- MathCAD.

, Excel - , - , MathCAD svds ( - . 2.2.1).

21 22

. 2.2.1. , Excel : ) MathCAD,

, ; ) (. . 2.2.2)

Excel ; ) (. . 2.2.3) -

: Create an empty Excel worksheet -

Excel; Create from file .

, Browse, - , , ( , - );

. 2.2.2. Excel

. 2.2.3.

23 24

) (. 2.2.4) Inputs = 0, Outputs = 1, Range , ( B3:D7), ;

. 2.2.4. ) MathCAD -

, : =, - ( ). - . . 2.2.5 MathCAD .

, Excel . , - . 2.2.1, :

B3:D7; 3:7. ) Out-

puts = 2, Range ( B3:D7, 3:7) . - (. . 2.2.6) ,

. 2.2.7 - MathCAD, - , .

. 2.2.5. MathCAD

. 2.2.6.

,

MathCAD ( submatrix), - MathCAD - Excel MathCAD Excel.

. , - . MathCAD - .

25 26

. 2.2.7. MathCAD 2. MathCAD ,

Excel (, ).

, MathCAD

: x 8,6,4,2= y 2,4,6,8= . 2:5, 3:D6.

: ) ) 1; ) Create an empty Excel

(. . 2.2.3); ) Outputs = 0, Inputs = 2

Starting Cell : - 2, 3 (. . 2.2.8) ;

) MathCAD - , - (. . 2.2.9), x, y - ;

) MathCAD . , Excel (-, ..) . 2.2.10 -

, - MathCAD.

. 2.2.8.

. 2.2.9.

27 28

. 2.2.10. , Excel MathCAD

,

MathCAD Excel MathCAD - .

2. MathCAD

MathCAD,

: , ( ). :

MathCAD ;

. ( -

) - - MathCAD, -.

( -) (, ) - , - ( -). MathCAD, - . , (7080 XX ) , - . MathCAD :

- ;

- - - ;

- MathCAD.

3.

MathCAD , -

, - MathCAD.

3.1.

. , - , MathCAD , .. .

3.1.1. : ax2 + bx + c = 0 :

29 30

2

1,24

2b b acx

a = . (3.1.1)

(3.1.1) ( ), MathCAD , . 3.1.1.

a 2:= b 5:= c 8:= d b2 4 a c:=x1

b d+2 a:= x2

b d2 a:=

x1 1.25 1.561i+= x2 1.25 1.561i=

a x12 b x1+ c+ 0.00000= a x22 b x2+ c+ 0.00000=

. 3.1.1. x1, x2 , -

1, 2 ( ) - [.] .

3.1.1. a, b, c,

-. .

, - MathCAD - .

3.2.

- . - ( ) .

3.2.1. y - x :

2 , 0;

, .

x xy

x =

(3.2.1)

x. . 3.2.1 - , - .

. 3.2.1. - (3.2.1)

: - ? (3.2.1) - -:

, ( , (3.2.1));

, - .

MathCAD : - , .

( ) :

, , , :

=>< ( - ).

.

2( )y x x= ( )y x x=

(1) (0) 0x

31 32

( . 3.2.2) -, . 3.2.1.

. 3.2.2. ! = -

. = .

3.2.1

< [ [>] [Ctrl] + [0] = [Ctrl] + [=] [Ctrl] + [3]

: 1

, 0 . - 1 , 0 .

3.2.1. x =3. - :

14);41);4) >+ xa .

, . 3.2.2.

3.2.2

(NOT)

(OR) (AND) (XOR)

. -

. 3.2.3.

3.2.3 NOT AND OR XOR

10 = 01 =

000 = 010 = 001 = 111 =

000 = 110 = 101 = 111 =

000 = 101 = 110 = 000 =

,

, . : 1 0; . , - : AND, OR XOR - .

3.2.2. - MathCAD, . 3.2.3.

x 0.2:= y 4:=x 3>( ) y 0>( ) =

x 3>( ) y 0

33 34

. 3.2.3. -

, if, :

if (, , ), if . - 1, .1, - . 2. - . 3.2.4.

. 3.2.4. - if

. 1 . 2 if (. 3.2.1).

3.2.1. if, - .

. >

=.0,;0,)(

2

xxxxxy

. 2

30, 1;( ) , 1 1;

30, 1.

xz x x x

x x

= < >

x 3:=y if x 0 x2, x,( ):= y 1.732=z x( ) if x 1 30, if 1 x 1 x, x2 30,( ),( ):=x 10 9.95, 10..:=

. 3.2.5.

, ,

. 3.2.5. z (x), .

MathCAD , - , , - -. :

ceil (x) , x; trunc (x) x; floor (x) , x; round (x, n) x -

n ; (x) 0 x < 0 1 -

; sign (x) ( 0 x = 0; -1, x 0);

.1 .2

. . 1 0

x

z(x)

35 36

signum (x) 1, x = 0 xx

.

3.3.

( ) -, , , , .

. -:

) ; ) .

, , , - .

3.3.1. z n , - :

)4(

1+= izi

, i = 1, , n; n = 10.

i , ( - iz ) n = 10 , .. - .

, .. , - - .

3.3.2. , - :

1 00.5 ( ), 0, 1, ... ; .n nn

ax x n x ax+

= + = = (3.3.1) 1nx + ,

: + axn2 1 , (3.3.2)

. - ,a (, = 9, 0.01 = ), - .

-. (- ), , , MathCAD , - .

, 3.3.1 . 3.3.1.

n 10:= i 1 10..:=zi

1i 4+:=

zT 1 2 3 4 5 6 7 8 9 101 0.2 0.167 0.143 0.125 0.111 0.1 0.091 0.083 0.077 0.071

=

. 3.3.1. z 3.3.1

. 3.3.2 ,

( ) :

,1 ; 1, ..., ; 1, ...,

1i j i n j m

i j= = =+ + . (3.3.3)

, : i , ; j , . , .

37 38

n 3:= m 6:=i 1 n..:= j 1 m..:=

Bi j,1

i j+ 1+:= B0.333

0.25

0.2

0.25

0.2

0.167

0.2

0.167

0.143

0.167

0.143

0.125

0.143

0.125

0.111

0.125

0.111

0.1

=

. 3.3.2. (3.3.3)

, . 3.3.1 3.3.2

ORIGIN=1.

. - until, :

until (,).

. 2 , . 1 0. . 1 , until 0, - . until .

. 3.3.3 , - 3.3.2.

3.3.1. MathCAD2001i

-- while.

ORIGIN 0:=a 9:= x0 a:= 10 6:=i 0 10..:=

xi 1+ until xi( )2 a xi

axi

+2

,

:=xT 9.0000 5.0000 3.4000 3.0235 3.0001 3.0000 0.0000( )=

. 3.3.3. until

, 7- x ( 6) - , 2( )ix a -, until 0. - , - :

nsolve last x( ) 1:= nsolve 5= xnsolve 3.0000000= , -

- - xn. - : xn () x n+1 ().

, . -, , , FORTRAN Pascal: , , .. - MathCAD, -.

39 40

4. -:

- .

4.1. -

- (-), , .. . - - -, (- ) -. - - , . 4.1.1.

. 4.1.1.

- MathCAD , . ( ) -.

- , - . - , - , .. . - , . - .

4.1.1. - -, , - -.

- -: , , - -.

- MathCAD. - :

- , - (. 4.1.1);

: : =; -

Add line (. . 4.1.1). , - - (. . 4.1.2);

f x( )

. 4.1.2. -

1 ( [Tab]) -. - , -

-

2

1

41 42

Add line (. . 4.1.3). . - [Delete];

f x( ) x x 2

z x

13

. 4.1.3. -

( 2), -

, - (. . 4.1.4).

f x( ) x x 2

z x

13

z . 4.1.4. -

x, - - (. ) z, -.

4.1.2. - - , - -.

. -

- - , :

< > < > .

! -

- ( : =) - .

4.2. - MathCAD

- - ( - ), ..:

< - > (< >) .

, - . .

, - , - . :

, , -, ;

, ;

, .

4.2.1. - , . 4.2.1. f(x), .

4.1.4, :

2 1

2

43 44

x 2:= f x( ) 1.587= f 3.23( ) 0.536 0.928i+=z f x 2.5+( ):= z 1.866=

, z -

z, -.

4.2.2. - - , - (. . 4.2.1).

x 2:=f x x 2+

z x

13

z

:=

f 1.587=

x 2= . 4.2.1. - -, - .

4.2.3. - , - .

5.

- MathCAD

- , - MathCAD.

5.1. -

, - , (. . 3.1). , , - - , .

5.1.1. - - ax2 + bx +c = 0

2 1/ 2

1,2( 4 )

2b b acx

a = m .

- root_poly2 . 5.5.1. - - . . , x1, x2 -, . [.] - . - - -.

root_poly2 a b, c,( ) d b2 4 a c

x1b d+2 a

x2b d2 a

x1

x2

:=

z root_poly2 12 7, 10,( ):= z 0.292 0.865i+0.292 0.865i

=

--

45 46

. 5.1.1.

5.2. -

, . ( ) .

-- MathCAD if. , - . if . 3.2, - if.

. - if - . - , . 5.2.1. if

. 5.2.1. if 2 (

). 1 1 ( , ), -, 1. = 0, 1 . - , .

- otherwise, , - 2, , - 0 (. . 5.2.2). otherwise if.

if

otherwise

. 5.2.2. 2 otherwise :

, if; otherwise

; otherwise -

2. 5.2.1. -, y(x), 3.2.1. - . 5.2.3.

y x( ) x2 x 0ifx otherwise

:=

y 2( ) 1.414= y 4( ) 16= . 5.2.3. 3.2.1 , , otherwise

, . - otherwise.

5.2.2. - - z(t)

z(t) =t t

t t

t t

3

2

34

3 4

, ;ln( ), ;

, .

< >

- . 5.2.4.

2 1

1 2

47 48

z t( ) t 3 t 3ift 2 3 t 4ifln t( ) otherwise

:=

z 8( ) 512= z 2( ) 4= z 7( ) 1.946= . 5.2.4. 5.2.2

, z(t) ln(t)

, , - -.

! 3 ln(t), -

- .

5.2.1. -, -

:

. y = 3

4

1 , 1;21 , 0 1;31 , 0.4

xx

x x

x x

49 50

poly n a, b, t,( )x a0y b0

n 0if

x a0 a1 t+y b0 b1 t+

n 1if

x 0y 0

otherwise

x

y

:=

. 5.2.7. 5.2.3 5.2.4. x, y. -,

x , y . . 5.2.8. v, x, y. - . 5.2.9.

arrangement x y,( )v0 xv1 y

x y>if

v0 yv1 x

otherwise

v

:=

. 5.2.8. 5.2.4

x 2:= y 8:= xy

arrangement x y,( ):= x 8= y 2=

. 5.2.9. - arrangement 5.2.2. a, b, c. -, -

. a b c , - , a b c> > , - , .

5.2.3. x, y. -, , .

5.2.4. a, b, c. -, n : n = 3, ; n = 2, ; n = 1, - .

5.3. -

, () - , . - , - . :

; . . 3.3. -

. for ( ). :

for . , - . 5.3.1;

51 52

for

. 5.3.1. for

1 , -

; 2 ,

; 3 , .

, (- ) Add line , , . 5.3.1. -, 3.3.1. - . 5.3.2. , ORIGIN ( - ) 1.

form_vec1 n( )

zi1

i2 1+

i 1 n..for

z

:=form_vec1 5( )

0.5

0.2

0.1

0.059

0.038

=

. 5.3.2. -

5.3.2. x -2 2 0.5 - f(x) = e-x. Cos (2x) y, .. y1 = f(-2), y2 = f(-1.5) ..

-

xk, x0 - , d . , (2(2))/0.5+1=9. -, y

9 . , - : x, - f(x); i, . - , , , x, .

- . 5.3.3. , . y, 1 .

form_vec2 x0 xk, d,( ) i 1yi e

x cos 2 x( )i i 1+

x x0 x0 d+, xk..for

y

:=

z form_vec2 0 0.8, 0.1,( ):=

zT 1 0.887 0.754 0.611 0.467 0.328 0.199 0.084 0.013( )=

. 5.3.3. 5.3.2 5.3.3. -, -

. . 5.3.4 -.

xk x

d

+

01,

1 2

3

53 54

. 5.3.4. - 5.3.3

5.3.1. - y

5.3.2, i. . - while. :

while . , - . 5.3.5;

while

. 5.3.5. while

1 ; 2 .

, - , .

while : - while, MathCAD . 1 (.. ),

, . - 0, .

5.3.4. -, , - 3.3.2. - . 5.3.6.

, - x0, x1, x2,... ( - until . . 3.3.3). - () ( xc) () xn.

, while - , .. . , - < 0, .

sqroot a ,( ) xn a

xc xn

xnxc

axc

+2

xn2 a >while

xn

:=

. 5.3.6. 5.3.4 MathCAD break, - break . break break ( ). break - if, ,

sum_vec_3 S 0

S S i+

i

2

5

7

for

S

:=

sum_vec_3 =

-

sqroot 9 0.1,( ) 3.00009=sqroot 9 0.0001,( ) 3.00000=

-

1 2

55 56

. if, - . - - break.

5.3.5. -, . - . 5.3.7.

10000. - , ierr - 1, -. xn, ierr - . ierr = 1, xn not solve - -. , - - ierr , , - , - .

sqroot_new a ,( ) xn aierr 1

ierr 0break

xn2 a

57 58

sum_conv a,( ) sum 0ierr 1

ierr 0break

a i( )

59 60

min_dist n m, A, c,( ) min 1010

min Ai j, cAmin Ai j,imin ijmin j

Ai j, c min

61 62

on error. - - :

< 1 > on error < 2 >.

. , . , . 5.3.11. on error - - angl(x,y). . 5.3.15.

angl x y,( ) x

y:= angl 2 0,( ) =angl 2 0,( )

angl x y,( ) 1010 xy

on error:=angl 2 0,( ) 1 1010= angl 2 2,( ) 1=

. 5.3.15. on error 5.3.11

error. -

:

error ("< >"). .

if, -.

5.3.12. -

v w. - . 5.3.16.

proj v w,( ) error "You cannot onto the 0 vector"( ) w 10 10

63 64

6.1.1. -, n . - . 6.1.1. - :

length(v) v; submatrix(A,i1,i2,j1,j2) , -

A, - i1 i2 j1 j2. ORIGIN = 0.

form_vec3 n v,( ) N length v( )v n Nifsubmatrix v 0, n 1, 0, 0,( ) n 1if otherwise

:=

x

2

5

7

9

2

1

:=

form_vec3 3 x,( )

2

5

7

=

form_vec3 10 x,( )

2

5

7

9

2

1

=

. 6.1.1. 6.1.1

6.1.2. -,

2 +1 1 (- . 6.1.2). - . 6.1.2. : mod(y,modul) - x modul;

ceil(y) , y. 6.1.3. -, -

A [ ],a b . - .

- :

f x a, b, A, B,( ) B A x a( ) x b( )( )+:= , 2, 4, 6, 2a b A B= = = = . 6.1.3. ( )x , < 0 1 .

0 2 4 6

1

0

1

1.5

1.5

f x( )

60 x

. 6.1.2. 6.1.2

2 0 2 4 63

1

1

3

55

3

f x 2, 4, 6, 2,( )

62 x

. 6.1.3. 6.1.3

-

f x( ) 1 mod ceil x( ) 2,( ) 1if1 mod ceil x( ) 2,( ) 0if

:=

x 0 0.05, 6..:=

65 66

6.1.4. MathCAD max polz x y= + , maxx x2 + 5.45x 8.12 = 0; poly - y2 + 25.3y 9.5 = 0. - . 6.1.4. - , . 5.1.1 (. 5.1.1).

X root_poly2 1 5.45, 8.12,( ):= X 1.218

6.668

=

Y root_poly2 1 25.3, 9.5,( ):= Y 0.3725.67

=

xmax if X0 X1> X0, X1,( ):= xmax 6.668=

ypol if Y0 0> Y0, Y1,( ):= ypol 0.37=

z xmax ypol+:= z 6.298=

. 6.1.4. 6.1.4

6.1.1. -,

, . 6.1.2, -: ) 1; ) 8. - (. 6.1.2).

6.1.2. -, -

, . 6.1.4, . - -.

6.1.3. -,

S n=1, 2, 3. - 6.1.4, , h, a, b, r, n -.

. 6.1.4.

6.2.

, . MathCAD ORIGIN=1. 1 ( , )

6.2.1. -, A :

,sin( ), ;sin cos , ;sin cos , .

i j

i j i ja i j i j

j i i j

+ == + > + ifai j, sin j( ) cos i( )+ i j

67 68

6.2.1. x y, n . -, B n n :

2

,2 , ;

, .i i i i

i ji i i i

x y x yB

x y x y

+ >= + >

6.2.2. x y, n -

. -, q n, i- 1, ( , )i ix y - r .

6.2.3. A n n . -, , 1, 0.

6.2.4. x, y, n . : xi yi - xi, yi.

2 (

, ) 6.2.2. y, n . - -, :

, ia y b ; .

- . 6.2.2. 6.2.3. n n .

-, , ,i ja C b . -, , - i j> . , : ) - ; ) - , , . -, . 6.2.3 .

calc_num n y, a, b,( ) k1 0k2 0

k1 k1 1+ a yi bifk2 k2 1+ yi 0>if

i 1 n..for

k1

k2

:=

. 6.2.2. 6.2.2

calc_num2 n C, a, b,( ) k 0

k k 1+ a Ci j, bifj 1 i 1..for

i 2 n..for

k

:=

. 6.2.3. 6.2.3

6.2.4.

cos ( )10iia i= + i=1,2,...,n; n=10.

-, - , [0.5, 1].

: ) - - , -; ) - . - . 6.2.4.

69 70

calc_posl_1 n a, a1, a2,( ) k1 0k2 0

c a i( )k1 k1 1+ c 0ifj 1 m..for

Di m, k

i 1 n..for

D

:=

. 6.2.5. 6.2.5

6.2.4. A, B n m . - -, , ,i j i jA B

6.2.5. D n m . -, i- -

i- D, ,i jD , .

3 ( , ) 6.2.6. y, n . - -, :

; , [ ],a b . - . 6.2.6. ,

sum, pr , -.

calc_sum1 n y, a, b,( ) sum 0pr 1

sum sum yi+ a yi bifpr pr yi yi 0>if

i 1 n..for

sum

pr

:=

. 6.2.6. 6.2.6 6.2.7. A n m . -

, i- . - . 6.2.7.

6.2.6. x, n . - -, .

6.2.7. x, n . - -, -, : -

a l( ) cosl

10l+:=

calc_posl_1 10 a, 0.5, 1,( ) 52

=

A

1

1

1

23

9

:=

calk_mat_1 3 2, A,( )1

1

1

1

2

0

=

71 72

, , .

sum_mat_2 n m, A,( )sum 0

sum sum Ai j,+j 1 m..for

Ai 1, sum

i 1 n..for

A

:=

. 6.2.7. 6.2.7

6.2.8. A n m . -, , i- - i- .

6.2.9. A n m . -, , ,i jA , -.

6.2.10. - n = 20 m = 5 . n m . -, , .

6.2.11. - n = 20 m = 5 . n m . -, .

4 ( ) 6.2.8. y, n . - -, -. - . 6.2.8.

min_max_vec n y,( ) min y1max y1imin 1imax 1

max yiimax i

yi max>if

min yiimin i

yi min

73 74

max_fun n x, f,( ) xmax x1imax 1fmax f xmax( )

fmax f xi( )imax ixmax xi

f xi( ) fmax>ifi 1 n..for

fmax

xmax

imax

:=

. 6.2.9. 6.2.9

min_mat_1 n m, A,( )jmin 1min Ai 1,

jmin jmin Ai j,

Ai j, min

75 76

F2 n( ) f 0 n 0if

f n

f f n 1( )n n 1break n 1if

1while

otherwise

f

:=

F2 5( ) 120= . 6.3.2. n 3.6.2. v, n . - -, , jv x , ,x .

- . 6.3.3. for, -, - search - (, 0 < ). - . 6.3.4. search v x, ,( ) j 1

j j 1+v j x >while

j

v j

:=

. 6.3.3. 6.3.2 search_1 v x, ,( )

break v j x ifj 1 last v( )..for

j

v j

:=

. 6.3.4. 6.3.2

- while ( ) . , . , . . 6.3.3 . 6.3.3.

2 0xx e = . (6.3.1) 510 = -

, [ ]4,0 , - ( ). , - , - . ( ) 0f x = -

( )x x= . (6.3.2) 0x -

[ ],a b , ( ) 0f x = ( [ ]4,0 ).

(6.3.2), 1 0( )x x= . 1x

(6.3.2) 2 1( )x x= . ,

1( )n nx x = (6.3.3) ( -). .

1. 0 1 2, , ,..., ,...nx x x x - x , .. , - , .. ( ) 0f x = .

j 1 500..:= w j ceil 1000 cos j( )( ):=

search w 400, 10 3,( ) 397400

=

search_1 w 400, 10 3,( ) 397400

=

77 78

2. 0 1 2, , ,..., ,...nx x x x , .. , . , . [ ],a b - ( )x x= , -

' ( ) 1x q < . (6.3.4)

( )a x b , (6.3.5) (6.3.3) , - [ ]0 ,x a b . , q (6.3.4), - 0 1 2, , ,..., ,...nx x x x x . - nx ,

1(1 )

n nqx x

q = , (6.3.6)

, -

nx x , (6.3.7) x . -

(6.3.2). 2x

x e= ( - - ), .. 2( )

xx e = . -

(6.3.4) , MathCAD (. . 6.3.5).

x( ) ex2:= d x( )

x x( )d

d:= x 4 3.95, 0..:=

4 3 2 1 00

0.2

0.4

0.6

0

d x( )0.5

04 x . 6.3.5. (6.3.4)

, q (6.3.4) 0.5, (6.3.4).

(6.3.5) ( )x [ ]4,0 (. . 6.3.6). , (6.3.5) [ ]4,0 ( )x - 4 ( ) 0x < < . :

[ ]0 4,0x (6.3.1).

(6.3.6):

10

6 1 0.5( )0.5

:= 1 10 6=

-, - . 6.3.7. - . ( )x - . (6.3.3) for ( . 5.3.4). , 0 , nx x , -, ,

79 80

( [ ] ).

4 3 2 1 01

0.5

00

1

x( )

04 x

. 6.3.6. (6.3.5)

posl_prib x 0 , ,( ) ierr 1x n x 0( )

ierr 0break

x n x 0 if

x 0 x n

n 1 10000..for

x n "not solve" ierr 1ifx n

:=

f x( ) x2 ex:= x( ) ex2:= x0 2:= 10 6:=

xapp posl_prib x0 , ,( ):=xapp 0.70347= f xapp( ) 3.01 10 7=

. 6.3.7.

6.3.1.

3( ) 5 20 3 0f x x x= + = .

- , - [ ]0,1 , . - while (. 5.3.3).

6.3.2. 3 2( ) 3 24 1 0f x x x x= + + = .

- , [7, 1] [2, 12], - 0.001. - while (. 5.3.3).

7. MathCAD

-

MathCAD.

7.1.

XX - , . - (, [10]). - :

- ;

- .

, , .

? - :

81 82

, ;

, - , .. ;

-. , - (, , - );

-, - .

MathCAD? - (. 4). - - :

- - . - ;

- - MathCAD -.

7.2.

MathCAD , :

, -; , -

. - -.

7.2.1. - -

:

b

adxxf )( , (7.2.1)

-, [a, b].

- Simpson (f,a,b,N) ( . 7.2.1) N, - Adapt-Simpson (f, a, b, ) , , . . :

,12

83 84

ORIGIN 0:=

Simpson f a, b, N,( ) h b aN

S f a( )( ) f b( )+

S S 4 f a i h+ h2

++

i 0 N 1..for

S S 2 f a i h+( )+i 1 N 1..for

h6

S

:=

Adapt_Simpson f a, b, ,( ) Int1 Simpson f a, b, 5,( )Int2 Simpson f a, b, 10,( )Int2 Int2 Int1

85 86

Reference:F:\COPY\__.mcd

. 7.3.1. -

7.3.1. -

7.2.1. - (

. 7.2.1) __ F:\Copy.

- , . 7.3.1, - -, (. . 7.3.2).

Reference:F:\COPY\__.mcd

f2 x( ) ex2 cos 2 x( ):=

Adapt_Simpson f2 0, 1, 10 8, 0.4290978=

. 7.3.2. - , : - ,

- MathCAD, - ;

- -. MathCAD, - -, - - .

7.4. MathCAD Internet

MathCAD - -, , Internet Web-,

MathCAD . - -, . Internet.

Web Library MathCAD , . :

Resource Center ( ) (. . 7.4.1);

. 7.4.1. Resource Center Web Library ; -

- (. . 7.4.2); (. . 7.4.3) -

.

Reference:F:\COPY\__.mcd

Reference:F:\COPY\__.mcd

87 88

. 7.4.2. Web Library MathSoft ( MathCAD) -

MathCAD (URL- http://www.mathcad.com/library/Mathcad_Files.asp). - . 7.4.4. . , Civil and Mechanical Engineering 94 (- 2002 ).

. 7.4.3. Web Library

89 90

. 7.4.4. MathCAD

, , (. . 7.4.5, Civil and Mechanical Engineering). MathCAD ( ).

. 7.4.5. Civil and Mechanical Engineering

MathCAD Exponenta (URL- www.exponenta.ru/soft/Mathcad). . 7.4.6 - MathCAD.

91 92

. 7.4.6. ,

Exponenta Web-, -

MathCAD, - . . 7.4.7 Web-. :

collab.mathsoft.com/~mathcad2000 MathCAD ( twt.mpei.ac.ru/ochkov/Collab/Collab.htm);

www.mathsoft.com/mathcad/library/3Dplots/ - MathCAD ( twt.mpei.ac.ru/ochkov/Lace/Lace.htm);

www.mathsoft.com/appsindex.html MathCAD;

www.mathsoft.com/mathcad/library/world.html - MathCAD (MathCAD Files Around the World);

www.mathsoft.com/books.html MathCAD; http://twt.mpei.ac.ru/ochkov/MC_ODE/Stiff_ODE/Stiff_DE.htm

( MathCAD);

http://petrsu.karelia.ru/psu/Deps/IMO/Complex/ - MathCAD - MathCAD;

http://www.keldysh.ru/comma - - ( .. ). -, MathCAD;

http://www.mpei.ac.ru/homepages/mm/ .

http://www2.latech.edu/~dmg/#Civil Engineering

. 7.4.7.

, -

MathSoft, MathCAD. - , ( ) MathCAD . - , , . - . - MathCAD.

: Resource Center (. . 7.4.1)

Collaboratory; Collabora-

tory (. . 7.4.7). .

- - . - (, MathCAD).

( MathCAD) . MathCAD File Send. -

93 94

Outlook Express. MathCAD .mcd. .

. 7.4.7.

3. -

MathCAD

, - . :

; , -

; . -

MathCAD, - (-). , , MathCAD.

8.

MathCAD

, , . , -.

8.1.

-, .

8.1.1.

: ( ) 0f x = . (8.1.1)

x , (8.1.1) , -, x - . , 2 2x x= :

95 96

1 22, 1x x= = . , .

( )f x x , - . . 8.1.1 2( ) 2f x x x= + , 2 2x x= , ( ) 0f x = ( ).

. 8.1.1.

( ), - , - :

11 1 0 0n nn na x a x a x a+ + + + = . (8.1.2)

1 1 0, , ..., ,n na a a a , , - ( ). - , .

, (8.1.2) n - n

1 2( ) ( ) ( ) ( ) 0nf x x x x x x x= = , (8.1.3) 1 2, , ..., nx x x (8.1.2). (8.1.2) ( )jx x ,

11 1 0

n nn n

j

a x a x a x ax x

+ + + + (8.1.4)

, jx . , (8.1.3). - .

, (, , ), -. sin( ) 0x = , - ( 0, 1, 2,...)x n n= = m m .

, : ( ), ( . 8.1.1), ( -). .

- :

; . .

8.1.2. ( ) 0f x =

[ ],a b , ( ) 0f x = . [ ],a b . , , , , -

97 98

. : .

-: ( )f x [ ],a b , [ ],a b

( ) 0f x = .

( )f x [ ],a b ( )f x x .

MathCAD, . .

8.1.1. 3 23 3 0x x+ = . (8.1.5)

. . 8.1.2 3 2( ) 3 3f x x x= + , -

MathCAD. , - : [ ]3, 2 , [ ]2, 1 , [ ]0, 2 . , -, .

. 8.1.2. (8.1.5)

8.1.2. 3 26 21 52 0x x x + + = . (8.1.6)

. . 8.1.3 ( )f x = 3 26 21 52x x x + + ,

MathCAD. , [ ]2, 0 . (8.1.6) . , .

. 8.1.3. (8.1.7)

8.1.3.

, - ( ,a b ). -,

b a , (8.1.7) . [11] , , , (- ) .

8.1.3. , -

(8.1.5). -.

,a b

99 100

, ( )f x -, .. ( ) ( ) 0f a f b < .

- root_2 . 8.1.4. - . .

root_2 a b, f, ,( ) a b+2

break b a

101 102

root: root [11], -

- ;

, root -, TOL. TOL , root -, . TOL , root , -. TOL MathCAD, : 0.01TOL = . - TOL , - - TOL ( 0.001);

, TOL, ;

f(x) -, root(f(x), x) r, . TOL. - f(x) = 0 - g(x) = 0,

)(

)()(xf

dxd

xfxg = .

8.1.5. root, 2 0xe ax = - 1 10 1.

. 8.1.6.

solve_root a x,( ) root e x a x2 x,( ):=a 1 10..:= x0 0:=xa solve_root a xa 1,( ):=xa

0.7040.54

0.459

0.408

0.371

0.344

0.322

0.304

0.289

0.276

=

0 5 100.2

0.4

0.6

xa

a

. 8.1.6.

polyroots. -

(8.1.2) n ( 5) - polyroots. poly-roots(v), v , n +1 , - , .. 0 0 1 1, , ..., n nv a v a v a= = = . .

8.1.6. polyroots, (8.1.6), (. 8.1.2). MathCAD, . 8.1.7.

8.1.1. root, -:

[ ]3sin( ) 0.35 3.8 0, 2,3x x x+ = ; [ ]3 14 0, 1,3x xx e e x + = .

103 104

v

a0

a1

a2

a3

:= polyroots v( )1.578

3.789 4.313i3.789 4.313i+

=

. 8.1.7. polyroots

Given. -

Given, - :

Given Find Minerr

(8.1.8)

, - , -. , .

Find , Find(x), x , -. , Minerr(x), - .

Find(x), Minerr(x) - - ( ), MathCAD. Find(x) - (. . 8.1.8) . , -

MathCAD, . - .

. 8.1.8. Find

Minerr(x). 8.1.1. -

Find(x), Minerr(x) Given , .

8.1.7. Given, - 8.1.2 [ ]5, 1 . - . 8.1.9.

8.1.2. Given, - :

[ ]1 ( ) 0, 0,1x tg x x = ; [ ]1 sin ln(1 ) 0, 0,2x x x x + + = ; [ ]3 4ln( ) 5 0, 2, 4x x x = .

105 106

. 8.1.9. Given

8.2.

, - MathCAD .

8.2.1.

- . - , . ,

2

2

5

3

x y

x y

+ = + = (8.2.1)

2, 1x y= = , ( ).

, -, :

; .

- ().

.

8.2.2. () -

:

1,1 1 1,2 2 1, 1

2,1 1 2,2 2 2, 2

,1 1 ,2 2 ,

... ;... ;

. . .... ,

m m

m m

n n n m m n

a x a x a x ba x a x a x b

a x a x a x b

+ + + =+ + + =

+ + + = (8.2.2)

n m . -

Ax b= , (8.2.3) A n m , b m . , : ) A -, .. n m= ; ) A , .. - .

b - n .

lsolve, : lsolve(,b), , b .

8.2.1. 1 2

2 3

1 2

2 12 4 40;7 4;

10 5 20.

x xx x

x x

+ =+ =+ =

(8.2.4)

lsolve, - .

MathCAD . 8.2.1. .

a3 1:= a2 6:= a1 21:= a0 52:=f x( ) a3 x

3 a2 x2+ a1 x+ a0+:=x 2:=

Given

f x( ) 0

4 x 1

Find x( ) 1.578=

107 108

A

2

0

10

12

1

5

47

0

:= b40

4

20

:=

x lsolve A b,( ):= A 810=

x

44

0

= A x b0.000000

0.000000

0.000000

=

A x b 0.000000000=

. 8.2.1. 8.2.1. lsolve,

:

1 2 3

1 2 3

1 2 3

100 6 2 200;6 200 10 600;

2 100 500.

x x x

+ = + = = (8.2.5)

1 2 3

1 2 3

1 2 3

10 12,2 10 13,

2 2 10 14.

x x

x

+ + = + + = + + = (8.2.6)

8.2.3. MathCAD -

, - MathCAD2001i 200.

- .

, - . , 8.2.2.

8.2.2. : 2;

8 3 .y xy x== + (8.2.7)

-. MathCAD, . 8.2.2, - , (8.2.7), .

x 3 2.95, 6..:=y1 x( ) x2:=

y2 x( ) 8 3 x+:=

5 0 5 1020

0

20

40

y1 x( )

y2 x( )

x

. 8.2.2. , :

(-2, 2), (5, 20).

- (.. ). - Given, - (8.1.8). -, , . -, .

Find - , Find(x), x , . - < >. .

: ;

109 110

, - ;

, Given Find ( Minerr).

, , Find, - . ,

f(a, b, c, ):= Find(x, y, z, ).

a, b, c,, .

(- ) , :

; , ,

-;

- . -;

, - . TOL. - - TOL ( - 0.001).

, - Minner . Find - .

8.2.3. Given, (8.2.7) 8.2.2. -. . 8.2.3.

sB4.702

22.105

=

sB Find x y,( ):=x 0> y x2y 8 3 x+

Giveny 20:=x 5:=

sA1.702

2.895

=

sA Find x y,( ):=y x2y 8 3 x+

Given

y 2:=x 2:=

. 8.2.3. 8.2.2

8.2.1. Given, :

( )sin 2 2,

cos 1 0.7.

x yy x+ = + =

( )

=+=+

.01cos2,4,0sin

xyxy

=+=++

.3,1)1sin(,8,0)1sin(

xyyx

=+

=.3,1)1sin(

,12)sin(xy

yx . 8.2.4. Minerr ,

111 112

2 2 2 2

0.95;

( 1) ( 1) 5.5.

x y

x y

+ =+ + + =

MathCAD . 8.2.4.

x 0:= y 1:=Given

x2 1+( )2 y2 1+( )2+ 5.5x y+ 0.95z Minerr x y,( ):=

z0.106

1.056

=

z0 z1+ 0.95=z0( )2 1+ 2 z1( )2 1+ 2+ 5.5=

. 8.2.4. 8.2.4

9.

MathCAD -

, ( ), . - - (, ), ( ). -, , ( -). , - , , .

9.1.

MathCAD: Maximize(f,) -

; Minimize(f,) -

, f , ; - ( ) , - .

! Maximize, Minimize ( ) - .

9.1.1. :

2 2 2( , , ) 10 2 36 4 3 18g x y z x x y y z z= + + + + . (9.1.1)

x, y, z, g(x, y, z) - .

MathCAD, , . 9.1.1.

g x y, z,( ) 10 x2 2 x 36+ y2+ 4 y+ 3 z2+ 18 z:=x 1:= y 1:= z 1:= ""

x

y

z

Minimize g x, y, z,( ):=x

y

z

1

23

= g x y, z,( ) 20=

. 9.1.1. (9.1.1) 9.1.2. :

32204163241

41),(

22 vvuuevuf += . (9.1.2)

113 114

u, v, f(u,v) - .

MathCAD, . 9.1.2. .

d u v,( ) 14 e

41 32 u 16 u2 4 v2 20 v+32:=

u 0:= v 0:= "" u

v

Maximize d u, v,( ):=

u

v

12.5

= d u v,( ) 0.0795775=

d u 0.01 u+ v 0.01 v+,( ) 0.0795673=d u 0.001 u v 0.001 v,( ) 0.0795774=

. 9.1.2. (9.1.2)

9.1.1. :

)zyx(sin))zy(cos)yx((cos)z,y,x( += . (9.1.3)

. 9.2. Maximize, Minimize,

Given (. (8.1.8)) , - .

9.2.1. :

ba50)ba(100)b,a(F 2 = (9.2.1)

.0a;1b;5b2a + (9.2.2)

a, b, (9.2.1) (9.2.2).

MathCAD, , . 9.2.1. , - (9.2.2).

F a b,( ) 100 a b( )2 50 ab:=

a 1:= b 1:=Givena 2 b+ 5 b 1 a 0a

b

Maximize F a, b,( ):=

a

b

0

2.5

= F a b,( ) 625=

a 2 b+ 5= b 2.5=

. 9.2.1. (9.2.1) 9.2.1.

-. ( ), Given, -, . Find, Minerr ( . 8.2.2).

9.2.2. :

22 )x1()xy(100)y,x(f += . (9.2.3)

, . -:

x9y;0y;0x . (9.2.4)

MathCAD . 9.2.2.

115 116

f x y,( ) 100 y x2( )2 1 x( )2+:=x 2:= y 3:=

Given

xf x y,( )d

d0

yf x y,( )d

d0

x 0 y 0 y 9 x x

y

Minerr x y,( ):=

x

y

1

1

=

f x y,( ) 3.538 10 8=

. 9.2.2. 9.2.3 ( ). -

100 1 2 3( , , )x x x 20 .

4, 3.4 2 , 340 , 4.75, 11 2 , 400 . , - 4, 3 2 . - .

MathCad, . 9.2.3.

x1 x2+ x3+ 100= 4.75 x1 11 x2+ 2 x3+ 534=

4 x1 3.4 x2+ 2 x3+ 340=

x1

x2

x3

56

20

24

=

x1

x2

x3

Maximize f x1, x2, x3,( ):=x1 x2+ x3+ 100

4.75 x1 11 x2+ 2 x3+ 7004 x1 3.4 x2+ 2 x3+ 340

x3 20x2 20x1 20Given

x3 1:=x2 1:=x1 1:=f x1 x2, x3,( ) 4 x1 3 x2+ 2 x3+:=

. 9.2.3.

9.2.4 ( ). v :

2

1 2 3( ) 2 2 .N v v v v v= + + :

3

1

1, 0.2, 1,3.i ii

v v i=

= = MathCAD, , . 9.2.4.

117 118

N v( ) v( )2 v2

12

+:=

v2 1:= v0

0

1

=Givenv 1 v 15>

v Minimize N v,( ):= v0.2

0.6

0.2

=

N v( ) 0.64= v 1=

. 9.2.4.

9.2.1 ( ). :

210 1592)( xxxxF ++= . -

:

.291029;25835.0

;4737;0;0;0

210

210

210

210

++

++

xxxxxx

xxxxxx

. : (0, 13, 8).

9.2.2 ( ). :

).12(2)20(5)15(),,( ++= vvuuvuQ -

:

.907;100423

;0;0;0

++++

vuvu

vu

: (7.5, 10, 6).

10. MathCAD

MathCAD - : , , .

10.1.

( -) - (, ) , .

MathCAD -. . 10.1.1 ( d), ( x , - ).

10.1.1 MathCAD

( )2

2

1 exp , 022

x >

( ), ,dnorm x ( ), ,qnorm ( ), ,vnorm m

!

x

ex

( x ,

0> )

( ),dpois x ( ),qpois ( ),rpois m

1

b a , [ ]bax , 0, [ ]bax ,

( ), ,dunif x a b ( ), ,qunif a b ( ), ,runif m a b

( ) xnxxn ppC 1 , nx 0

( ), ,dbinom x n p ( ), ,qbinom n p ( ), ,rbinom m n p

119 120

. 10.1.1 MathCAD

2 0, 0x ;

( ) 12/2/ 2221

nn xen

x , 0>x

( > 0n )

( ),dchisq x n ( ),qchisq n ( ),rchisq m n

( )( ) 21

22

11

2

++

+

n

nx

nn

n

( > 0n )

( ),dt x n ( ),qt n ( ),rt m n

MathCAD -. x ( )p x , - :

( )x p x dx

= . . 10.1.1

MathCAD ( q ) .

MathCAD . , , - , -.

. 10.1.1 ( r ), . m .

( )rnd x , [0, 1].

10.1.1. . 10.1.1 MathCAD, : N ( - 20, 100); 2 - ( 10). - 100.

MathCAD - . : ( ), -, .. - - m . .

N rnorm 100 20, 10,( ):= i 1 100..:= rchisq 100 10,( ):=

0 50 10060

40

20

0

20

40

Ni i

i

mean N( ) 21.505=var N( ) 95.484=

mean ( ) 10.312=var ( ) 25.356=corr N ,( ) 0.014=cvar N ,( ) 0.704=

. 10.1.1. . 10.1.2 , -

.

121 122

,X Y m , X Y .

10.1.2

MathCAD

X ( )mean X

X ( )var X

X ( )side X X ( )median X X ( )mode X ,X Y ( ),cvar X Y

10.1.2. . 10.1.1

MathCAD, , 10.1.1.

MathCAD ( ). .

, { }, 1,i i N= ( N ). L+1 1 2 1L Lz z z z +< < <

123 124

L, - { }kz (10.1.7), (10.1.8);

X N, { }ix .

{ }, 1, , .kn k L= K histogram(int,X): int (L+1),

, 1 , , 1.kz k L= +K int , - L, - { }kz (10.1.7), (10.1.8);

N, { }ix .

- 2L , kd ( [ ]1, , 1, , ,k kz z k L+ = K kn .

, his-togram, kd - , , .k k kn w p

10.1.3.

, ,N 10.1.1. N = 1000.

. 10.1.2 ,N . 10.1.3 - histogram L = 11. kd , solidbar ( , ). - , kx d= .

ORIGIN 1:= N 1000:= N rnorm N 20, 10,( ):=L round 1 3.222 log N( )+( ):= L 11=HN histogram L N,( ):= dN HN 1 := nN HN 2 :=h dN2

dN1:= h 5.909= pN

1N h nN:= k 1 L..:=

60 40 20 0 200

0.02

0.04

pNk

dnorm dNk20, 10,

dNk

. 10.1.2.

ORIGIN 1:= N 1000:= rchisq N 10,( ):=L round 1 3.222 log N( )+( ):= L 11=

H histogram L ,( ):= d H 1 := n H 2 :=h d 2 d 1:= h 3.545= p

1N h n:= k 1 L..:=

0 10 20 30 400

0.05

0.1

p kdchisq d k 10,

d k

. 10.1.3. 2

125 126

10.2. . -

- . -, x - y

( )y f x= . ( )f x , - ,x y { }, , 1, ...,i ix y i n=% , ( )i i iy f x = +% ,

i , . -

S(x), - f(x).

. 1. , S(x).

, S(x) S(x)=0+1, 0, 1 . , -. ( { }, , 1,...,i ix y i n=% ) . , , - .

2. S(x) . F(a). - , :

2

1

( ) ( ( , )) ,n

iii

F a y S a x=

= % (10.2.1) S(, x) S(x) {aj}, j=1, 2,, M.

MathCAD -

. MathCAD -

S(x). . 10.2.1, , {xi}, i=1, , n, Y , { }, 1, ..., .iy i n=%

10.2.1

slope(X, Y) 1

0 1( )S x a a x= + intercept(X, Y)

0 0 1( )S x a a x= +

regress(X,Y,m)

0 1, ,...,T

ma a a -

0 1( )m

mS x a a x a x= + + +L m ( 5m ). , (. 10.2.2)

line(X, Y) 0, 1

0 1( )S x a a x= +

linfit (X, Y,)

1 2, , .., ma a a - 1 1 2 2( ) ( ) ( )m ma x a x a x + + +L )(),...,(1 xx n - - (x), linfit

genfit(X,Y,,F)

1 2, , ..., ma a a S(x). F(x,a) - (m+1), S(x)

( ) , 1,..., ,j

S x j ma

= m, -

,ja 1,...,j m= (. 10.2.3) 10.2.1. -

x { }ja ,

127 128

regress, interp(v, X, Y, x), v -, regress (. 10.2.2). - - (. 10.2.1, 10.2.3).

10.2.2. linfit genfit - , 1,, m linfit , genfit - , ( ) - .

10.2.1. ( ) 2 0.5f x x= + 1n + { }ix

. . 10.2.1.

A2.0757

0.4218

=

2 1 0 1 21

2

3

4

yi

y iS xi( )

xi

S x( ) A0 A1 x+:=A line x y,( ):=y

0.139=y y +:= rnorm n 1+ 0, max y( ),( ):=yi a0 a1 xi+:=xi 2 0.2 i+:=

0.1:=i 0 n..:= a1 0.5:=a0 2:=n 20:=ORIGIN 0:=

. 10.2.1. 10.2.2. ,

. 10.2.2, 0 1( )

mm mS x a a x a x= + + +L m = 2 m = 3.

, - . 10.2.2. , m = 3 -, .. { }iy% .

ORIGIN 0:= i 0 5..:=

x

1

2

3

4

5

6

:= y

0.8

3.5

8

15

19

15

:= v2 regress x y, 2,( ):=coef2 submatrix v2 3, 5, 0, 0,( ):= coef2

8.157.995

0.634

=S2 x1( ) interp v2 x, y, x1,( ):=

v3 regress x y, 3,( ):= S3 x1( ) interp v3 x, y, x1,( ):=coef3 submatrix v3 3, 6, 0, 0,( ):=

0 2 4 610

0

10

20

yi

S2 xi( )S3 xi( )

xi

coef3

7.133

11.2315.734

0.606

=

. 10.2.2.

10.2.3. , . 10.2.3 (- X Y ),

20 1 2( ) exp( )S x a a x a x= + + .

MathCAD, . 10.2.3, genfit. - ( , )x a , - , -

129 130

0 1 2, ,a a a . - iy% .

X

0.4

1

1.4

2

4

:= Y

11.2

5

3

6

2

:= x a,( )

ea0 a1 x+ a2 x2+

ea0 a1 x+ a2 x2+

ea0 a1 x+ a2 x2+ x

ea0 a1 x+ a2 x2+ x2

:=

a0

1

0

1

:= A genfit X Y, a0, ,( ):= S x( ) x A,( )0:=i 0 4..:= z 0.2 0.25, 4.25..:=

0 1 2 3 4 50

5

10

15

Yi

S z( )

Xi z,

A

2.82

1.1940.178

=

. 10.2.3.

- , . 10.2.1, , . . 10.2.2 .

10.2.2

expfit(X,Y,ao)

1, 2, 3 - 21 3( )

a xS x a e a= + . (- 3) , .. 1, 2, 3

lgsfit(X,Y,ao)

1, 2, 3

xaeaxS

32

1

1)( += . ( 3) -

1, 2, 3

lnfit(X,Y) 1, 2

1 2( ) ln( )S x a x a= + logfit(X,Y, ao)

1, 2, 3

1 2 3( ) ln( )S x a x a a= + + . ( 3) 1, 2, 3

pwrfit(X,Y,ao)

21 3( )

S x a = + . ( 3) 1, 2, 3

sinfit(X,Y,ao)

1 2 3( ) ( )S x a Sin x a a= + + . ( 3) 1, 2, 3

10.2.4. , . 10.2.4, -

2

1 3( )a xS x a e a= + .

- MathCAD, . 10.2.4.

131 132

a0

4

1

1

:= ORIGIN 0:=

Y

6.3

1.8

11.6

29.1

27.3

40.5

50.4

96.7

:= X

0

1

2

3

4

5

6

7

:= A expfit X Y, a0,( ):=S x( ) A0 e

A1 x A2+:=i 0 7..:= z 0 0.05, 7.5..:=

0 2 4 6 80

50

100

150

Yi

S z( )

X i z,

A

3.756

0.453

3.019

=

. 10.2.4.

. 10.2.1 10.2.2 - , -, . - - , - , . .

MathCAD loess(X,Y,d), . X, Y , 0d > ( 0.75 0.85). d, - . d loess(X,Y,d) regress(X,Y,2). - loess(X,Y,d) v,

interp(v,X,Y,z), z.

10.2.5. . 10.2.5 MathCAD, - loess(X,Y,d)

0.8d = 3d = . , 3d = - .

2 1 0 1 21

0

1

2

3

yi

y iS1 xi( )S2 xi( )

xi

S2 x1( ) interp v2 x, y, x1,( ):=S1 x1( ) interp v1 x, y, x1,( ):=v2 loess x y, d2,( ):=v1 loess x y, d1,( ):=d2 3.2:=d1 0.4:=

y

0.152=y y +:= rnorm n 1+ 0, max y( ),( ):=yi a0 a1 xi+ a2 xi( )2+:=xi 2 0.2 i+:=

0.1:=i 0 n..:= a2 0.3:=a1 0.5:=a0 2:=

n 20:=ORIGIN 0:=

. 10.2.5.

. , . . MathCAD -

133 134

, - smooth (). :

medsmooth(Y,m) Y, n - - m (m m n< ). - , n - Y;

supsmooth(X,Y) Y, n -, ix , - X. - k ix k . X ;

ksmooth(X,Y,b) Y, n -, ix , - X.

1

1

( )

( )

ni j

jj

i ni j

j

x xh y

by x x

hb

=

=

=

%,

2

2

1( ) exp2 (0.37)2 0.37

th t =

.

, iy jy%

, b , - jy% .

10.2.6. . 10.2.6 MathCAD, - ( 38%) ksmooth b ( b =0.3 0.161 b =0.9 0.141).

"" i 0 n..:= xi i 0.1:= yi f xi( ):= ""

0.2:= rnorm n 1+ 0, max y( ),( ):= y y +:= ksmooth

Y1 ksmooth x y, 3 0.1,( ):=Y2 ksmooth x y, 9 0.1,( ):= Y1 yy = Y2 yy =

yi

y iY1i

Y2i

xi

. 10.2.6. ksmooth

0 1 2 3 4 5

0.8 0.6 0.4 0.2

0 -0.2

135 136

-

MathCAD . , - . - , - -. [ 2, 3, 9,10 ].

( ), MathCAD - - .

1. .. . / .. . .: - , 2001. 296 .

2. .. MathCAD2000: / .. . : , 2000. 596 .

3. .. MathCAD 8 Professional , Internet / .. , .. . .: , 1999. 512 .

4. .. Matlab: / .. . : , 2001. 560 .

5. .. Matlab 5.x.: 2- . . , 1999. 672 .

6. .. Mathematica 4: / .. . : , 2000 482 .

7. .. Maple V R3/R4/R5 / .. . .: , 1998. 381 .

8. .. - MathCAD. / .. , .. . : - , 1999. 32 .

9. .. MathCAD2001 / .. . .: -, 2002. 459 .

10. .. MathCAD / .. . .: - , 2001. 196 .

11. .. / .. , .. . .: , 1970. 486 .