Fortran 95

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X ELATEX. .

6 2018. https://www.materials.uoc.gr/el/undergrad/courses/ETY114/notes.pdf

stamatis@uoc.grhttp://creativecommons.org/licenses/by-nc-sa/4.0/deed.elhttp://creativecommons.org/licenses/by-nc-sa/4.0/deed.elhttps://www.materials.uoc.gr/el/undergrad/courses/ETY114/notes.pdf

i

1 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.6 . . . . . . . . . . . . . . . . . . . . . . . . . . 51.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 72.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

i

ii

2.6.4 . . . . . . . . . . . 172.7 Fortran 95 . . . . . . . . . . . . . . . . 18

2.7.1 . . . . . . . . 192.8 / . . . . . . . . . . . . . . . . . . . . . . . . . . 202.9 . . . . . . . . . . . . . . . . . . 21

2.9.1 . . . . . . . . . . . . . . . . . . . . . . 242.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.10.1 STOP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 313.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.1 IF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.2 SELECT CASE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 414.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 DO . . . . . . . . . . . . . . . . 41

4.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2.3 DO . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 DO . . . . . . . . . . . . . . . 484.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.1 EXIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.4.2 CYCLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.4.3 DO . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 575.1 . . . . . . . . . . . . . . . . . . . 57

5.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.1.2 . . . . . . . . . . . . . . . . . . . . . . . . 585.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.4 / . . . . . . . . . . . . . 635.5 . . . . . . . . . . . . . . . . . . . . 655.6 / . . . . . . . . . . . . . . . 685.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.8 . . . . . . . . . . . . . . . . . . . . . . . . 72

5.8.1 WHERE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

iii

5.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6 776.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

7 / 837.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837.2 FORMAT . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

7.2.1 FORMAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

8 998.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 998.2 . . . . . . . . . . . . . . . . . . . . . . . 1008.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

8.3.1 . . . . . . . . . . . . . . . . . . . . . . 1078.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088.6 RETURN . . . . . . . . . . . . . . . . . . . 1108.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

8.7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118.7.2 . . . . . . . . . . . . . . . . . . . . . . . 1128.7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1168.9 (recursive) . . . . . . . . . . . . . . . . . 1178.10 (ELEMENTAL) . . . . . . . . . . . . . 1198.11 MODULE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1218.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

141.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 141.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.1.3 hash . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.1 Bubble sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.2 Insertion sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.3 Quick sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2.4 Merge sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

iv

149.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

153

155

1

.

. -, , . , , .

, , . , .

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

1

2

, , , - . , , .

- . , , - .

- Fortran 95 . .

Fortran 95 - . Fortran 95 .

1.1

- : . , , :

1. [ ].

2. [ ] [- ].

3. [ ] .

. , (RAM) , .

, Fortran, , , . :

0. (, ).

Fortran 95 - . :

3

PROGRAM diplasio

IMPLICIT NONE

! 0INTEGER :: a, b

! 1READ *, a

! 2b = 2 * a

! 3PRINT *, "Todiplasioeinai"PRINT *, b

END PROGRAM diplasio

:

1.2

( ) - PROGRAMEND PROGRAM :

PROGRAM diplasio

IMPLICIT NONE.........END PROGRAM diplasio

diplasio. . .

/ 132 .

PROGRAM ... . ( ) IMPLICIT NONE. , , .

4

END PROGRAM .... Fortran 95

.

1.3

compiler ! . - ( ASCII).

! - , . - , , , . , , .

1.4

(variables) (, ) . , compiler . .

INTEGER :: a, b

. , . compiler (INTEGER) a b.

1.5

READ *, a

- a. * ,

5

, READ. .

. (2.2) ().

1.6

b = 2*a

: , : a - 2. *. , .

1.7

PRINT *, "Todiplasioeinai"PRINT *, b

: - ( ) b ( ).

- .

1.8

1. , .1

1 compiler - .

6

2

2.1

Fortran

INTEGER , ,

REAL DOUBLE PRECISION , ,

COMPLEX , ,

LOGICAL , ,

CHARACTER , .

2.1.1

.. abc :

INTEGER :: abc

..value1, value2, :

REAL :: value1REAL :: value2

, ,

7

8

REAL :: value1, value2

, compilers, 6 1.

, REAL . . Fortran 95 ( 2.7), . Fortran 77.

.. ab, :

DOUBLE PRECISION :: ab

-, 15 .

( ) :

COMPLEX :: z

.

2.1.2

, , , , , , . az, 0 9 _. 31 . . . _ .

- .

2.1.3

, ., READ... a

1 , , .

9

INTEGER :: a.....READ *, a

, , ,

INTEGER :: a.....READ (*,*) a

(2.2),

= [ ]

..

REAL :: a, ba = 5.2b = 3.0 + 2 * a

( ). ..

INTEGER :: k = 3

k 32.

.

2.2

= [ ]

, . , , -, , .

, 5.6.

2 , 8.8.

10

2.3

2.3.1

, - . + ...,

3, 12, 123456

2.3.2

, ( ) ( 6 ). . E, , 10 E :

2.034, 0.23, .44, 23., 2e 4 ( 0.0002), 2.3E2 ( 230.0) .

, ( ) D 10 ( 15 ). ..,

2d 4 ( 0.0002d0), 2.3D2 ( 230.0d0) .

Fortran - 3. - . , .. , .

E D . - . , 3.2 , Fortran 3.2D0 . , - . . -, 15 16 , 0 15 16 . , - = 3.14159265358979323846 . . . Fortran 3.141593 ( , ) 3.14159265358979 D0.

3, , , .

11

2.3.3

Fortran , , . . .., 3.1 + 2i

(3.1, 2.0),

i

(0.0, 1.0).

2.4

, , , PARAMETER . () ( , , , , ., ). .. ,

DOUBLE PRECISION, PARAMETER :: pi = 3.14159265358d0

2.5

+,, , / , , .

+,, -, , . / ,, , . MOD()

INTEGER :: a, b, p, y

a = 5b = 3p = a / b ! : p 1y = MOD(a,b) ! : y 2

. .. x3 Fortran x**3.

+, , , / ( ), .

12

, . . ..

INTEGER :: i, j, k

i = 2 + 3 ! i 5j = 2 * 3 ! j 6i = i + j ! i 11i = 2 * i + j ! i 28i = 17 / j ! i 2k = i * j / 3 + 2**3 ! k 12

. ..

INTEGER :: i, j, k

i = 2j = 6k = 2 * i + j ! k 10k = 2 * (i + j) ! k 16k = j * i / 3 ! k 4k = j * (i / 3) ! k 0

: +,, , / - (.. ), - . , , - , . , . ..

INTEGER :: i, jREAL :: x, y

i = 8.3 ! i 8j = 2y = 3.0

x = 4.0 / 3 ! x 4.0/3.0 x 1.33333

x = j + y ! x 2 + 3.0 x 2.0 + 3.0 x 5.0

i = 4 / 3.0 ! i 4.0/3.0 i 1.33333 i 1

x = 4 / 3 ! x 1 x 1.0

13

x = 2.0 * 4 / 3 ! x 8.0/3 x 8.0/3.0! x 2.66667

x = 4 / 3 * 2.0 ! x 1 2.0 x 2.0

: (.. ) . ; .

PRINT *, 0.1d0-0.3d0+0.2d0PRINT *, 0.1d0+0.2d0-0.3d0

2.6

Fortran, INTEGER , REAL, DOUBLE PRECISION .

2.6.1

- , Fortran CHARACTER . .. ch :

CHARACTER :: ch

() ():

ch = 'a'ch = 'D'ch = "R"

, . , "Thisisamessage" 17 , , CHARACTER . -

CHARACTER (17) :: ch

ch = "Thisisamessage"

14

2.6.2

(LOGICAL) (.. /, /,). , .. a, :LOGICAL :: a

.TRUE. .FALSE.. .. - .TRUE. a a = .TRUE.

2.6.3

Fortran COMPLEX - . COMPLEX :: z

, compiler, , , . Fortran 77 . Fortran 95 (2.7) . , Fortran, DOUBLE COMPLEX .

Fortran

, , , (,) . - :

COMPLEX :: z

z = (1.2, 4.56) ! z 1.2 + i4.56

- 0

COMPLEX :: z

z = 2 ! z 2.0 + i0.0z = 2.3 ! z 2.3 + i0.0

( , ):

15

COMPLEX :: z, z1, z2

z1 = 3 ! z1 3.0 + i0.0z2 = (4.6, -4.2) ! z2 4.6 i4.2z = z1 + z2 + 5.1 ! z 12.7 i4.2

.

. ..

COMPLEX :: z

READ *, z

(1.6, -3.8)

( ).

CMPLX().

COMPLEX :: zREAL :: x, y

x = 3.4y = -9.1

z = CMPLX(x,y) ! z 3.4 i9.1

z = (x,y)

, .

+,-,*,/,** .

z1 = + i, z2 = + i,

z1 z2 = ( ) + i( + ) .

16

* Fortran. /.

2.2, 2.5, ( ), , . , . - , :

COMPLEX :: zREAL :: x

z = (2.3,4.5) ! z 2.3 + i4.5x = z ! x 2.3

, - :

COMPLEX :: zINTEGER :: i

z = (2.3,4.5) ! z 2.3 + i4.5i = z ! i 2

.

/

REAL() ( ) :

COMPLEX :: zREAL :: x

z = (1.2,-8.7) ! z 1.2 i8.7x = REAL(z) ! x 1.2

AIMAG():

COMPLEX :: zREAL :: x

z = (1.2,-8.7) ! z 1.2 i8.7x = AIMAG(z) ! x 8.7

17

CONJG():

COMPLEX :: z1, z2

z1 = (1.2,-8.7) ! z1 1.2 i8.7z2 = CONJG(z1) ! z2 1.2 + i8.7

() |z| ABS():

COMPLEX :: zREAL :: norm

z = (1.2,-8.7) ! z 1.2 i8.7norm = ABS(z) ! norm

1.22 + 8.72

2.6.4

. ..

DOUBLE PRECISION :: xINTEGER :: i, j

i = 3j = 2

x = i/j ! x 1.0d0

x, , (1.5) .

DOUBLE PRECISION :: xINTEGER :: i, j

i = 3j = 2

x = DBLE(i)/DBLE(j) ! x 1.5d0

. . .

18

Fortran : REAL(), DBLE(),INT(), CMPLX(). (INTEGER , REAL, DOUBLE PRECISION , COMPLEX) REAL, DOUBLE PRECISION , INTEGER , COMPLEX . - .

, -, x = 1.0d0 * i / j

x = (i+0.0d0) / j

x (;).

2.7 Fortran 95

Fortran , , , . Fortran 95 - , . Fortran 77 REAL DOUBLE PRECISION . Fortran 95 , . .

Fortran 95 :REAL (prc) :: a

, a (kind)prc. , . , .

, prc , :

SELECTED_REAL_KIND (). - ( ) ( ). ..INTEGER, PARAMETER :: prc = SELECTED_REAL_KIND(12)

prc 12 .

Fortran 95 19

KIND(). .

INTEGER, PARAMETER :: prc = KIND(1.0D0)

prc -.

: -, - . , - REAL . :

INTEGER, PARAMETER :: prc = SELECTED_REAL_KIND(12)

REAL (prc) :: c

INTEGER, PARAMETER :: prc = KIND(1.0D0)

REAL (prc) :: b

b 1.0D0, . , c 12 . , prc , .

. - :

INTEGER, PARAMETER :: prc = KIND(1.0D0)

COMPLEX (prc) :: z

z .

2.7.1

Fortran 95 2.3 ., 340 3.4E2 340.0 3.4D2 . , prc,

20

340.0_prc

, ,

3.4E2_prc

:

INTEGER, PARAMETER :: prc = KIND(1.0D0)

REAL (prc) :: a, bREAL (prc), PARAMETER :: pi = 3.141592653589793_prc

a = 2.4_prcb = a + 5.6E2_prc

, , , - , .

2.8 /

a , , READ:

READ *, a

, ,

READ (*,*) a

PRINT

PRINT *, a

, , WRITE:

WRITE (*,*) a

-, . ..

INTEGER :: aDOUBLE PRECISION :: b

a = 4b = 4.5d0

PRINT *, "Oitimeseinai", a, b

21

* READ(*,*)/WRITE(*,*) . . * - . READ */PRINT * , , * ( ).

2.9

(AIMAG(), CONJG(), CMPLX()) 2.6.3, Fortran . - 2.1.

- () ., DOUBLE PRECISION - DOUBLE PRECISION .

Fortran :

PROGRAM riza

IMPLICIT NONEDOUBLE PRECISION :: x

PRINT *, "Dwseenapragmatiko"READ (*,*) x

PRINT *, "Htetragwnikhrizaeinai"WRITE (*,*) SQRT(x)

END PROGRAM riza

PROGRAM imitono

IMPLICIT NONEDOUBLE PRECISION :: theta, x

22

2.1:

Fortran

95(

).

ABS(x)

.

.

ABS(x)

( xx).

.CEILING(x)

O

.

.

FLOOR(x)O

-

.

.

INT(x)

|x|,

x.

,

-

(

).

NINT(x)

.

,

.

AINT(x)

|x|,

x.

.

ANINT(x)

x>0

AINT(x+0.5)

AINT(x-0.5).

.

SQRT(x)

.

-

.

EXP(x)

(ex).

LOG(x)

(lnx).

(,

].LOG10(x)

(logx).

.

MOD(x,y)

x/y,

x-INT(x/y)*y.

x,y

,

.

y=

0.MODULO(x,y)

R

[0,y)

x=Qy+R

Q

.

x,y,

y=0.

MODULO(x,y)

x-

FLOOR(x/y)*y.

x,y,

y=

0.

23

2.1:

Fo

rtran95

(

).

MA

X(x1

,x2,

...)

-

.

,

-

.

MIN(

x1,x

2,...)

-

.

,

-

.

DIM(

x,y)

MA

X(x-

y,0.0)

.

,

-

.

REAL

(x)

.

.

DBLE

(x)

.

.

CONJ

G(x)

.

.AI

MAG(

x)

.

.CO

S(x)

.

rad.

SIN(

x)

.

rad.

TAN(

x)

.

rad.

ACOS

(x)

.

[1

:1],

[0

:]

rad.

ASIN

(x)

.

[1

:1],

[/2:/2]

rad.

ATAN

(x)

.

[/2:/2]

rad.

ATAN

2(y,

x)

tan1(y/x

).

-

.

(

:]

rad.

COSH

(x)

.

SINH

(x)

.TA

NH(x

)

.

24

PRINT *, "Dwseenapragmatiko"READ (*,*) theta

x = SIN(theta)

PRINT *, "Toimitonoeinai"WRITE (*,*) x

END PROGRAM imitono

( ), , 2.1.

(rad).

2.9.1

RANDOM_NUMBER () - . , - [0, 1). 8 CALL . , :

PROGRAM randIMPLICIT NONEDOUBLE PRECISION :: x, y

CALL RANDOM_NUMBER(x) ! to x einai tyxaios metaksy 0, 1CALL RANDOM_NUMBER(y) ! to y einai tyxaios metaksy 0, 1

END PROGRAM rand

RANDOM_NUMBER () RANDOM_SEED (). - .

[a, b) . r [0, 1) x = r+ , x . x = (b a)r + a a x b.

25

RANDOM_NUMBER () elemental, 8.10, . N N .

2.10

, PROGRAMEND PROGRAM :

PROGRAM onoma

IMPLICIT NONE.........

END PROGRAM onoma

( onoma) , - . - . Fortran 95, . , 132 , - , &. ..

INTEGER :: a, b

, ,

INTEGER :: a, &b

, & ( ).

PROGRAM onoma . ( ) IMPLICIT NONE. , , .

, modules, 8. .

26

2.10.1 STOP

, - END PROGRAM . , STOP . . STOP , , 5 . , , STOP. STOP( ) .

27

2.11

1. . ( copy-paste!) athroisi.f90, .PROGRAM athroisiIMPLICIT NONE

INTEGER :: a, bINTEGER :: c

PRINT *, "Dosedyoakeraious"READ *, a,b

c = a + b

PRINT *, "Toathroismatouseinai:"PRINT *, c

END PROGRAM athroisi

2. , , .

3. ( ).

4. 2.3167895443, 4 103, 102, 3/2. . ;

5. , x,y,z:

d = x2 + y2 + z2

d = x2/y + z d = 2.45(x+ 1.5) + 3.1(y + 0.4) + 5.2 z/2 d = (12.8x+ 5y)/(11.3y + 4z) d = x2/3 + y2/3 + z2/3

x = 1.5, y = 2.5, z = 3.5 :20.75, 4.4, 19.79, 0.7502958579881658, 5.457604592453865.

28

6. :

() ,() ,() ( -

),() ( -

).

= 2R E = R2.

7. a, b, c a2+b2 = c2 . m, n m > n, a = m2 n2, b = 2mn, c = m2 + n2.4 .

8. .: 10. ;

9. - , , . : 200000, : 2 days, 7 hours, 33 min & 20 seconds.

10. Gauss ( 2099) :

.

:() r1 = 19.() r2 = 4.() r3 = 7.() ra = 19r1 + 16.() r4 = ra 30.() rb = 2(r2 + 2r3 + 3r4).

4 .

29

() r5 = rb 7.() rc = r4 + r5 + 3.

rc , .5

. - rc, .

11.

() ,() ,() .

12. . 80 cm. - . . :

() ,() cm,() km, m cm.

, 1253 : 100 240 cm 1 km, 2m, 40 cm. (, 2001)

13.

= 4 tan1(1) ,

= 8 tan1(1/3) + 4 tan1(1/7) .

-, x ( )

tan1(x) = cos1(1/1 + x2

)= sin1

(x/1 + x2

).

= 3.14159265358979323846264338 . . .. - ;

14. - , (, ), , , .

5 rc 30, (rc 30) .

30

3

3.1

Fortran ( - ) ( 3.1). -

== /= > < >= 2.0 .TRUE. 2/=1+1 .FALSE.. - . , , ==,/= .

, - , 2.5, .

: .

Fortran .NOT., .AND., .OR., .EQV., .NEQV.. , , :

31

32

.NOT. :

.NOT. (4 > 3) .FALSE. .NOT. (4 < 3) .TRUE.

.AND. .TRUE. .TRUE.. .FALSE.:

(4 > 3) .AND. (3.0 > 2.0) .TRUE. (4 < 3) .AND. (3.0 > 2.0) .FALSE.

.OR. .TRUE. .TRUE., .FALSE.:

(4 > 3) .OR. (3.0 < 2.0) .TRUE. (4 < 3) .OR. (3.0 < 2.0) .FALSE.

.EQV. .TRUE. .FALSE.:

(4 < 3) .EQV. (3.0 < 2.0) .TRUE. (4 < 3) .EQV. (3.0 > 2.0) .FALSE.

.NEQV. .TRUE. .FALSE.:

(4 < 3) .NEQV. (3.0 < 2.0) .FALSE. (4 < 3) .NEQV. (3.0 > 2.0) .TRUE.

, .EQV. .NEQV., - . - ==, /= .

Fortran LOGICAL :

LOGICAL :: a, bINTEGER :: i, max

...

a = 3==2b = ( (i > 0) .AND. (i < max) )

33

3.2

3.2.1 IF

IF -. () -, . Fortran :

IF (condition) THEN...... // block A

ELSE...... // block B

END IF

(condition), , . , block ELSE END IF ( ).

, , ...

IF (val > 0.0d0) THENPRINT *, "Oarithmoseinaithetikos"max = val

ELSEPRINT *, "Oarithmosdeneinaithetikos"

END IF

IF. () . .

ELSE , ELSE :

IF (condition) THEN......

END IF

ELSE IF,

IF (condition) THENcommand

END IF

34

:

IF (condition) command

THEN, END IF. IF IF, ..

IF (cond1) THEN...

ELSEIF (cond2) THEN

...ELSE

...END IF

END IF

IF (cond1) THEN...

ELSE IF (cond2) THEN...

ELSE...

END IF

, ELSE IF:

IF (cond1) THEN...

ELSE IF (cond2) THEN...

ELSE IF (cond3) THEN...

ELSE IF (cond4) THEN...

ELSE...

END IF

IF, cond1, cond2, - . , . , ELSE ( ) IF.

35

3.2.2 SELECT CASE

SELECT CASE , IF. :

SELECT CASE (i)CASE (value1)...CASE (value2).........CASE (valueN)...CASE default...

END SELECT

i ( ) ( ), . value1, value2, ,valueN ( -).

, i value1,value2,,valueN. , - CASE. , END SELECT . value1, value2,,valueN default , ., END SELECT .

CASE .

- . 0 , 1 , 2 , 3, .

INTEGER :: iCHARACTER (5) :: direction

READ *, i

SELECT CASE (i)CASE (0)

direction = "up"CASE (1)

36

direction = "down"CASE (2)

direction = "left"CASE (3)

direction = "right"CASE default

direction = "none"END SELECT

PRINT *, "Hkateythinsieinai", direction

i, - . .., (0, 4, 8) (1, 5, 9), (2, 6) (3, 7).

SELECT CASE (i)CASE (0,4,8)

direction = "up"CASE (1,5,9)

direction = "down"CASE (2,6)

direction = "left"CASE (3,7)

direction = "right"CASE default

direction = "none"END SELECT

, ,

:

, CASE

CASE (0:9)...

[0, 9]. , () Fortran. ..

SELECT CASE (i)CASE (:-1)... ! entoles otan i < 0CASE (0)... ! entoles otan i == 0

37

CASE (1:)... ! entoles otan i > 0

END SELECT

38

3.3

1. , .: 2 0.

2. - , . , .

3. ax = b, a, b . a, b, : () a = 0, () a = 0 b = 0, () a = 0 b = 0.

4. ax2 + bx + c = 0, () a, b, c . a,b,c, . (, ), , .

5. .

6. / 5 .

7. - 2015 :

( ) 0 - 20000 22%

20000,01 - 30000 29%30000,01 - 40000 37%

40000,01 45% , 48000 , 20000 22% ( 4400 ), 10000 29% ( 2900 ), - 10000 37% ( 3700 ) 8000 45% ( 3600 ). 14600 . ( ) .

39

8. :

, , , , , , 31 .

, , , 30 . 28 ,

29. 4,

. 400. : 4 100 . 400.

16 - 1923 ( ) 1 1923 ( )., 1923 15 .

, SELECT CASE, .

9. Zeller (, , ) : d (1, 2, 3, . . . , 31), m (1, 2, . . . , 12) y . 1 () 2 () m 12 1 y. ,

() a 13(m+ 1) 5.() j, k ,

100.() b j 4.() c k 4.() h a, b, c, d, k

j.

h 7 : 0 , 1 , . .

40

4

4.1

, . DO Fortran , - . :

, , ,

, .

4.2 DO

: 1, 2, 3, 4, 5, . , ,

WRITE (*,*) 1WRITE (*,*) 2WRITE (*,*) 3WRITE (*,*) 4WRITE (*,*) 5

. , , .. 500;

41

42

. ,

INTEGER :: i

i=1WRITE (*,*) i

i=2WRITE (*,*) i

i=3WRITE (*,*) i

i=4WRITE (*,*) i

i=5WRITE (*,*) i

, , , 1. DO

INTEGER :: iDO i = 1,5

WRITE (*,*) iEND DO

, .. 500 : i.

4.2.1

DO

DO = , , ........END DO

. , , , .

DO 43

( (,) ) 1. , 0.

:

1. .

2. :

, .

, .

, - .

3. 2 , END DO, .

4. 2 , (, ) DO, END DO.

5. 2.

( ) ,

( )

, (.. EXIT, 4.4.1) .

( ) , , , , . , , . , DO END DO.

- 1 9 s. ,

44

INTEGER :: ss = 1 + 3 + 5 + 7 + 9

. , , :INTEGER :: ss = 0s = s + 1s = s + 3s = s + 5s = s + 7s = s + 9

, , -, s, .

, :INTEGER :: sINTEGER :: is = 0

i = 1s = s + i

i = i + 2s = s + i

i = i + 2s = s + i

i = i + 2s = s + i

i = i + 2s = s + i

:INTEGER :: sINTEGER :: is = 0DO i = 1, 9, 2

s = s + iEND DO

DO 45

4.2.2

, -,

= +

( , ).

, - DO.

100 , 1 100, - :INTEGER :: sum, isum = 0DO i = 1, 100

sum = sum + iEND DO

[0, 1000] - :INTEGER :: sum, isum = 0DO i = 0, 1000, 2

sum = sum + iEND DO

99, 97, 95,, 3, 1, , INTEGER :: iDO i = 99, 1, -2

WRITE (*,*) iEND DO

0.0, 0.1, 0.2, , 99.9, 100.0 INTEGER :: iDO i = 0, 1000

WRITE (*,*) 0.1d0 * iEND DO

46

( )

DOUBLE PRECISION :: xDO x = 0.0d0, 100.0d0, 0.1d0 !

WRITE (*,*) xEND DO

[5 : 108]

INTEGER :: k, ik = 0DO i = 5, 108

IF (MOD(i,2) == 0) k = k + 1END DO

, k - . 0 . - . , . .

0,1,1,2,3,5,8,13,, , , , - Fibonacci. n , n2 . , . . - : a , b c . c=a+b. : - . . , - , : ,

DO 47

, - . .

INTEGER :: a,b,c,ka = 0b = 1PRINT *, aPRINT *, b

DO k=2,nc = a+bPRINT *, ca = bb = c

END DO

, .

4.2.3 DO

DO -. :

, , 1, 2, 3, 4, 5.

WRITE (*,*) 1,2,3,4,5

,

WRITE (*,*) (i, i=1,5)

DO -, () .

(, = , , )

=, DO. 1. , . , DO .

1, 3, 5, 7, 9

48

WRITE (*,*) (i, i=1,9,2)

WRITE (*,*) (2*i+1, i=0,4)

DO , , 5.

4.3 DO

- . .. - , . , . . Fortran 95 DO . :DO........END DO

DO, DO END DO . , . Fortran 95 , , EXIT (4.4.1) .

4.4

4.4.1 EXIT

EXIT DOEND DO - . , . END DO.

: , DOUBLE PRECISION :: xDO

READ (*,*) xIF (x > 0.0d0) EXIT

END DO

, .

49

4.4.2 CYCLE

CYCLE DOEND DO. -, - DO , , . - DO . DO , 4.3, .

: 10 , -.

INTEGER :: iDOUBLE PRECISION :: xDO i=1,10

READ (*,*) xIF (x < 0.0d0) CYCLE ! x

WRITE (*,*) "Htetragwnikhrizaeinai", SQRT(x)END DO

4.4.3 DO

DO. ( :) DO, , END DO , :

onoma: DO i=1, 10....

END DO onoma

, 2.1.2. DO EXIT CYCLE.

, , -. , , , ,

CYCLE onoma

EXIT onoma

onoma. ,

50

. EXIT .

outer: DO i=1,10DO j=1,10

DO k=1,20......IF (k > i+j) EXIT outer

END DOEND DO

END DO outer

, EXIT ( ) END DO outer.

51

4.5

1. 0.0, 0.1, 0.2,0.3, 0.4, , 2.5.

2. 20 2 (20, 21, ...,219).

3. . n! :

n! = 1 2 3 . . . n, 0! = 1.

n 12.

4. 53 [3.5, 6.5] (- ).: n [a, b] , h. x1 = a, x2 = a + h, x3 = a + 2h, , xn =a+ (n 1)h. xn b, h = (b a)/(n 1).

5. 30 - f(x) = x(x2 + 5 sin(x)) [5, 5]. .

6. 1000 . H , 1.3%. 15 .

7. , 2011 10815197. 0.53% , 15000000.

8. () a,b. - 1. , - a b:

a < b . b = 0 a . b > 0

b a b.

135 680.: 5

1http://en.wikipedia.org/wiki/Euclidean_algorithm

http://en.wikipedia.org/wiki/Euclidean_algorithm

52

9. . , . (.. 6 , 1, 2, 3, 6 1+2+3+6 = 12 = 26).

10. .

11. N Fibonacci2:

fi+2 = fi+1 + fi, i 0, f0 = 0, f1 = 1 .

: 0, 1, 1, 2, 3, 5, 8, . . .. . n . 0 n < 31.

12. , . I: -: 1 . II: ( ) , 1 . - .

13. . . .

14. Gauss 10 28,

() ,() ,() ,() 18 ,

1930 .2http://oeis.org/A000045

http://oeis.org/A000045

53

15. 8 39 - (, ).

16. 8/2/2015. 8 39 17000 .

17. . (n!!)

n!! =

1 3 5 (n 2) n n ,2 4 6 (n 2) n n ,

1 n 0.

n 15.

18.

ex

ex =n=0

xn

n!,

sinx

sinx =k=0

(1)kx2k+1

(2k + 1)!,

cosx

cosx =k=0

(1)kx2k

(2k)!.

- . - 1010.

19. Gauss, a

2a = n(n+ 1) +m(m+ 1) + k(k + 1)

n,m,k . - a,

54

n,m,k . n,m,k (- n m k). 16 ([0, 1, 5] [0, 3, 4] [2, 2, 4]), 104 ([2, 4, 13] ), 111 ([1, 10, 10] [0, 9, 11] ).

20. - Lagrange3. , ( ) -.: , . - , . 0 . .

21. n=1

[12

n2cos

(9

n +(n + 3)(n 3)

)]=

2

e3

: ( 0). - 107.

22. : . () 2 - . . /- 0. . , .

23. 4 ( p, p+6, p+12 ) . 10000.

3http://mathworld.wolfram.com/LagrangesFour-SquareTheorem.html4http://oeis.org/A047948

http://mathworld.wolfram.com/LagrangesFour-SquareTheorem.htmlhttp://oeis.org/A047948

55

24. 1 , 2 , 5 , 10 , 20, 50 , 100 (= 1) 200 (= 2). -. 300 ;: , 300 , 150 , 60 5 . 300 .: 471363

25. 120 - [100, 100]. RANDOM_NUMBER () (2.9.1). - .

26.

= 3 + 2k=1

k(5k + 3)(2k 1)!k!2k1(3k + 2)!

.

106 106. .: - .

27. 65728 ( ) :

65728 = 123 + 403 = 313 + 333 .

64232:

64232 = 173 + 393 = 263 + 363 .

k k = i3 + j3 ( ) i, j. i, j, k i j < k.

28. -:

3 = 1 + 2 12

56

5 = 3 + 2 12

9 = 1 + 2 22 = 7 + 2 12

15 = 7 + 2 22

27 = 19 + 2 22... =

...

.: 17, 137, 227, 977, 1187

29. n - 1 n . .. 3142 . .: 1 .: 4231

30. Goldbach. Goldbach . Goldbach, 4 Goldbach. [6, 10000].

5

, , , , . , - ( ) ( ), - ( ) TYPE 6.

5.1

5.1.1

- . , ; 30 365 , - . H Fortran, , - , . , 30 30 . .. temperature

DOUBLE PRECISION :: temperature(30)

, ,

DOUBLE PRECISION, DIMENSION (30) :: temperature

57

58

, , - . ..

DOUBLE PRECISION :: a, b(3), c(10)

a, 3 , b, 10 , c, . , , :

INTEGER, DIMENSION (10) :: a, b, c, d(4)

, (a,b, c) 10 (d) 4.

, . ,

PARAMETER (2.4):

INTEGER, PARAMETER :: n = 25DOUBLE PRECISION :: a(n)

:

INTEGER, PARAMETER :: n = 25DOUBLE PRECISION :: a(2*n*n-n/2+13)

5.1.2

( - ) , 1 1. , temperature (1), temperature (2), temperature (30). . .., o 3 temperature

DOUBLE PRECISION :: mo3

mo3 = (temperature(1) + temperature(2) + temperature(3)) / 3d0

5.1.3

( )...

1 .

59

DOUBLE PRECISION :: a(5)

a(1) = 3.4d0READ (*,*) a(2)

:

(/ /)2 . , 5 c, 3.0, 4.5, 2.0, 100.0, 0.99 ,

DOUBLE PRECISION :: c(5)

c = (/ 3.0d0, 4.5d0, 2.0d0, -1.0d2, 0.99d0 /)

( , ). ..

DOUBLE PRECISION, DIMENSION (5) :: c, d

c = (/ 3.0d0, 4.5d0, 2.0d0, -1.0d2, 0.99d0 /)d = c

, READ :

DOUBLE PRECISION :: c(5)

READ (*,*) c ! 5

- (RESHAPE(), SPREAD(), CSHIFT(), EOSHIFT(), .). .

5.1.4

, , 1. .. 6 2 (20,21, 22,,25) 0 5. . ..

2 [ ] Fortran 2003.

60

INTEGER :: powersOf2(0:5)

powersOf2 = (/1,2,4,8,16,32/)

, 6 , powersOf2(0),powersOf2(1), , powersOf2(5). ,

DOUBLE PRECISION :: a(M:N)

, ,

DOUBLE PRECISION, DIMENSION (M:N) :: a

M,N NM, a NM+1 , a(M), a(M+1), , a(N).

powersOf2, - [ , (/, /)] :

INTEGER :: a(0:5)INTEGER :: i

a = (/ (2**i,i=0,5) /)

DO (4.2.3) . powersOf2 a , a, , ( 20 ;).

5.2

, - 2 3 , .. . Fortran 95 , , ( 7 ). 10 20

DOUBLE PRECISION :: a(10,20)

, ,

DOUBLE PRECISION, DIMENSION (10,20) :: a

i j a(i,j). , - 3.

3 , , , , ., a, 23, a(1,1), a(2,1),a(1,2), a(2,2), a(3,1), a(3,2) .

61

.

(C)1 5.02 7.53 6.4...

...157 19.1158 21.4

......

364 4.5365 7.0

1 365.

, . , ( ) . , , :

DOUBLE PRECISION :: tempr(365)

,

1 2 . . . 14 15 . . . 30 31

1 5.0 7.5 . . . 8.3 9.2 . . . 12.3 11.02 4.5 6.5 . . . 7.0 9.0 . . ....

......

......

......

6 25.0 27.5 . . . 28.3 29.2 . . . 27.37 26.0 27.0 . . . 26.0 31.0 . . . 32.5 33.0...

......

......

......

11 5.0 6.5 . . . 7.6 10.0 . . . 11.012 5.0 7.5 . . . 8.5 9.5 . . . 12.0 12.5

.

: () ()., 12 31,

DOUBLE PRECISION :: tempr(12,31)

62

-. , 14 tempr(4,14)4.

5.3

, , PARAMETER , 2.4, - ( ) :

DOUBLE PRECISION, PARAMETER :: c(3) = (/2.0d0,2.5d0,4.0d0/)

( ) . ..,

DOUBLE PRECISION :: a(10)INTEGER :: i

DO i = 1,10READ (*,*) a(i)

END DO

, . . , a 5 8

DO i=1,5DO j=1,8

READ (*,*) a(i,j)END DO

END DO

( i j, i j, .) - , .

4 ,

.. (2, 30) (6, 31).

/ 63

IMPLICIT NONEDOUBLE PRECISION :: b1, b2, b3INTEGER :: i

DO i = 1, 3READ (*,*) bi ! , bi

END DO

b1, b2, b3 , . .

5.4 /

, . Fortran 95 , , , . .. , . , .

DOUBLE PRECISION, ALLOCATABLE :: a(:)

a. . N , N , ,

ALLOCATE(a(N))

-

DEALLOCATE(a)

, 5.

, a(M) a(N) M,N - ,

ALLOCATE(a(M:N))

(, , )

5 ( 8), ( , 8.8) .

64

DOUBLE PRECISION, ALLOCATABLE :: b(:,:)

ALLOCATE(b(M,N))

M,N . DEALLOCATE(b)

. ALLOCATE/DEALLOCATE -

:PROGRAM alloc

IMPLICIT NONEDOUBLE PRECISION, DIMENSION (:), ALLOCATABLE :: a, bDOUBLE PRECISION, DIMENSION (:,:), ALLOCATABLE :: cINTEGER :: M,N, K, P

READ (*,*) M, N, K, P

ALLOCATE (a(M), b(N), c(K,P))

! a, b, c! a, b, c

DEALLOCATE(a)DEALLOCATE(b, c)

END PROGRAM alloc

- ( ALLOCATE) ( DEALLOCATE ). -, ( ).

ALLOCATABLE , ALLOCATED() . .

: - :INTEGER :: nDOUBLE PRECISION :: a(n) ! . n;READ (*,*) n

,

65

INTEGER :: nREAD (*,*) nDOUBLE PRECISION :: a(n) !

, -, .

5.5

Fortran 95 . :

SIZE(): , , - . Fortran 95 SIZE() . ,

DOUBLE PRECISION, DIMENSION(10,10,30) :: a

SIZE(a), SIZE(a,1), SIZE(a,2) SIZE(a,3) .

LBOUND()/UBOUND(): . a LBOUND(a,1), LBOUND(a,2), . UBOUND(a,1),UBOUND(a,2), .

SUM(): SUM() - 6. ..

INTEGER :: a(10), s

a = (/ 1, 5, 4, 2, 12, 9, 8, 2, -4, -7 /)s = SUM(a) ! s 32

SUM() , :

INTEGER :: a(10), s, i

6 , , , + .

66

a = (/ 1, 5, 4, 2, 12, 9, 8, 2, -4, -7 /)s = 0DO i = 1, SIZE(a)

s = s + a(i)END DO

PRODUCT(): PRODUCT() 7. ..INTEGER :: a(10), p

a = (/ 1, 5, 4, 2, 12, 9, 8, 2, -4, -7 /)p = PRODUCT(a) ! p 1935360

p = 1DO i = LBOUND(a,1), UBOUND(a,1)

p = p * a(i)END DO

DOT_PRODUCT (): DOT_PRODUCT () - . DOT_PRODUCT(a,b)

i aibi a .

i ai bi a .

a,b LOGICAL , DOT_PRODUCT(a,b) .TRUE. .TRUE., .FALSE..

MATMUL(): MATMUL() - . (). , - . . , C = AB A, B, C , C=MATMUL(A,B). , M N B N P , C M P

Cij =Nk=1

aikbkj , 1 i M , 1 j P .

7 , , , *

67

C=MATMUL(A,B) , , ,

INTEGER :: i, j, kDOUBLE PRECISION :: s

DO i = LBOUND(A,1), UBOUND(A,1)DO j = LBOUND(B,1), UBOUND(B,1)

s = 0.0d0DO k = LBOUND(A,2), UBOUND(A,2)

s = s + a(i,k) * b(k,j)END DOC(i,j) = s

END DOEND DO

TRANSPOSE(): TRANSPOSE() .

DOUBLE PRECISION :: a(10,20), at(20,10)

at = TRANSPOSE(a)

at (i, j) (j, i) a.

COUNT(): COUNT() - , , , -. .. COUNT(A > 10) ( ) 10.

MAXVAL()/MINVAL(): MAXVAL() MINVAL() , , .

MAXLOC()/MINLOC(): MAXLOC() MINLOC() / . .. a (2, 3) , MINLOC(a) 2 ( ), 3 ( ). , . 1, ( ).

68

ALL(): ALL() , -, . , .TRUE.. .FALSE..

ANY(): ANY() , -, . - , .TRUE.. .FALSE..

- .

5.6 /

Fortran 90 - Fortran 95 , - (elemental).

=,+,, , /, 2.1 ( ). Fortran 95 . ,

DOUBLE PRECISION :: x

x = 3.4d0

DOUBLE PRECISION :: a(10)INTEGER :: i

DO i=1,SIZE(a)a(i) = 3.4d0

END DO

,

DOUBLE PRECISION :: a(10)

a = 3.4d0

/ 69

= a.

5.1.3. a = b, a, b , b a.

. , DOUBLE PRECISION, DIMENSION (10) :: a, b, cINTEGER :: i...... ! a,b

DO i=1,SIZE(a)c(i) = a(i) + b(i)

END DO DOUBLE PRECISION, DIMENSION (10) :: a, b, c...... ! a,b

c = a + b , DO. , DOUBLE PRECISION, DIMENSION (10) :: a, b, c...... ! a,b

c = a * bDOUBLE PRECISION, DIMENSION (10) :: a, b, cINTEGER :: i...... ! a,b

DO i=1,SIZE(a)c(i) = a(i) * b(i)

END DO .

, , - .

70

DOUBLE PRECISION :: lambda, a(10), b(20)

a = lambda * a

b = lambda * b

: lambda a,b, . ,

a = a * b

. , -

. ..

DOUBLE PRECISION :: x, y..... ! x

y = SQRT(x)

x y. a b:

DOUBLE PRECISION :: a(20), b(20)..... ! a

b = SQRT(a)

, ,

DOUBLE PRECISION :: a(20), b(20)INTEGER :: i..... ! a

DO i=1,SIZE(a)b(i) = SQRT(a(i))

END DO

, , - ( , 5.7), .

71

5.7

Fortran ... a 10 , 3, 4, 5, 6, 7 - a(3:7) ( 5 ) SIZE(a(3:7)) 5. - 5 . ,

DOUBLE PRECISION :: a(10), b(20)

a = b(1:10)

( 10 b a), a=b . a a(1:SIZE(a)) a ( 1).

, - , . , , , a 4 . a(1:7:2) . , . , a ,

a( : : )

, 1. . , .

a(:10) a(1:10)a(4:) a(4:SIZE(a))a(:) a(1:SIZE(a)) a

: .

DOUBLE PRECISION :: a(100), tempINTEGER :: i

DO i = 1, SIZE(a)/2temp = a(SIZE(a) - i + 1)a(SIZE(a) - i + 1) = a(i)a(i) = temp

END DO

72

a(i), a(SIZE(a)-i+1). ,

a = a(SIZE(a):1:-1)

a a.

. a 1020,

INTEGER :: a(10,20)

a(1:5,1:6) 56 5 6 .

(.. ) (.. ) a a(3,:) a(:,5) . .

5.8

H Fortran 95 . -, , . .

5.8.1 WHERE

WHERE IF, , . , WHERE

WHERE (condition)........

ELSEWHERE........

END WHERE

condition, , , . WHERE . ,

73

ELSEWHERE . , ELSEWHERE . .., a b, b a .

DO i = LBOUND(a,1),UBOUND(a,1)IF (a(i) > 0.0d0) THEN

b(i) = a(i)ELSE

b(i) = 0.0d0END IF

END DO

WHERE (a > 0.0d0)b = a

ELSEWHEREb = 0.0d0

END WHERE

, ,

b = MAX(a, 0.0d0)

WHERE, , ELSEWHERE . , , , WHERE. , a b

WHERE (a>=0.0d0) b = SQRT(a)

b a .

74

5.9

1. 100 . - i (i = 1, . . . , 100) i2 + 3i + 1. , .

2. N . j (j = 1, . . . , N ) sin(j/N). , 0.4.

3. A, B - 2030. (i, j) (i = 1, . . . , 20, j = 1, . . . , 30) A (i+ j)/3 B(i, j) 2i j/3.

() BT B .

() 8 A,BT - .

() ( ) A BT .

4. a, 100 . j (j =1, . . . , 100) cos(j/100). , 50 50 , , a(1) a(51), a(2) a(52),, a(50) a(100).

5. 0 12. , ex

ex x0/0! + x1/1! + x2/2! + + x12/12! .

x . EXP().

6.

1

=

n=0

((2n)!)3(42n+ 5)

(n!)6163n+1,

8 AMN , BNP Aij , Bij , CMp

Cij =

Nk=1

AikBkj .

75

. 15 3.1415926535898.: .

7. - 1000 : 2 ( ). .

8. , . n . : 1, 2, 3, . . . , n. . - ( ) ( ). n ( - ). i - i. . n, ;

9. M N M = 20, N = 60, K = 400 1 0. 1 (2.9.1). , 'x' 'o' .

10. 9 9 . , ( , , ). . . , , , . ( ) . , ( ) . .

76

. . .

6

, , , - . , , . , .

, . .. - , , , - , . , , ( , , , , .), ( , , , , .), . - . , 500 3000

INTEGER, PARAMETER :: N = 500 ! INTEGER, PARAMETER :: M = 3000 ! CHARACTER(30) :: studentName(N)CHARACTER(50) :: bookAuthor(M)CHARACTER(60) :: bookTitle(M)INTEGER :: studentAM(N), studentYear(N), bookYear(M)

77

78

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

, - -. : , - !

Fortran 95 , . TYPE,

TYPE onoma ......

END TYPE onoma

TYPE MODULE (8.11) TYPEEND TYPE . TYPE. , .

:

TYPE (onoma) :: x

. , -.

, - , , :

TYPE student

79

CHARACTER(30) :: studentNameINTEGER :: studentAM, studentYear

END TYPE student

TYPE bookCHARACTER(50) :: bookAuthorCHARACTER(60) :: bookTitleINTEGER :: bookYear

END TYPE book

. ,

TYPE studentCHARACTER(30) :: nameINTEGER :: am, year

END TYPE student

TYPE bookCHARACTER(50) :: authorCHARACTER(60) :: titleINTEGER :: year

END TYPE book

.

TYPE (student) :: sTYPE (book) :: bINTEGER, PARAMETER :: N = 500 ! INTEGER, PARAMETER :: M = 3000 ! TYPE (student) :: students(N)TYPE (book) :: books(M)

, s b, - student book.

, %. ,

TYPE (book) :: b

b%title = "Fortran95/2003explained"b%author = "MichaelMetcalf,JohnReid,MalcolmCohen"b%year = 2004

80

, . .., PRINT *, b%titlePRINT *, b%author IF (b%year > 2003) THEN

.....END IF

, . , , TYPE (book) :: b, cb%title = "Fortran95/2003explained"b%author = "MichaelMetcalf,JohnReid,MalcolmCohen"b%year = 2004c = b - :TYPE (book) :: bb = book("MichaelMetcalf,JohnReid,MalcolmCohen", &

"Fortran95/2003explained", 2004) - , . .

, .

, , , , READ :TYPE (book) :: bREAD *, b

, , , .

, .., 70 (, , ) .

81

INTEGER, PARAMETER :: N = 70INTEGER, DIMENSION (N) :: day, month, yearDOUBLE PRECISION :: temperature(N)

, . ( !)

INTEGER, PARAMETER :: N = 70INTEGER, DIMENSION (3,N) :: dateDOUBLE PRECISION :: temperature(N)

date date(1,j) j , date(2,j) date(3,j) ( , !). - (date) (temperature),, , -: date(1,5), date(2,5), date(3,5), date(1,8) - .

, - (date):

TYPE dateINTEGER :: day, month, year

END TYPE date

, (measurement):

TYPE measurementTYPE (date) :: dDOUBLE PRECISION :: temperature

END TYPE measurement

,

TYPE (measurement) :: meas(70)

j

meas(j)%d%day = 21meas(j)%d%month = 3meas(j)%d%year = 2014meas(j)%temperature = 24.5D0

82

6.1

1. - . , , , - , , , . .

7

/

7.1

, READ/WRITE ( PRINT) , , - . , - , , . / . -/ , 2.6.1, FORMAT . FORMAT .

FORMAT

READ (*, "(......)") a, b, iWRITE (*, "(......)") f, g, hhhh

, ,

READ "(......)", a, b, iPRINT "(.......)", f, g, hhhh

, * READ(*,*)/WRITE(*,*) * READ */PRINT *, , . -. , .

83

84 /

, - FORMAT . FORMAT * /. FORMAT - , module , END CONTAINS module. END CONTAINS . FORMAT , READ/WRITE , - ( , ,, module) .

READ (*, 12) a, b, i12 FORMAT (......)

WRITE (*, 25) f, g, hhhh25 FORMAT (.......)

READ 12, a, b, i12 FORMAT (......)

PRINT 25, f, g, hhhh25 FORMAT (.......)

FORMAT - .

7.2 FORMAT

FORMAT READ/WRITE.

, - FORMAT , , , .

,

I - w, . Iw.

WRITE (*, "(I5)") 12

FORMAT 85

12

- . , , . , w *. - . ,

-345

, format 7, :

READ (*, "(I7)") i

, n, format Iw.n. n , 0. ,

WRITE (*, "(I5.3)") 99

099

( - ) E, EN, ES, F -, w, d ( ) . Ew.d, ENw.d, ESw.d, Fw.d. w - , . d -. FORMAT .: F . ,

WRITE (*,"(F9.2)") -146.29d0

-146.29

86 /

, 0, format w 0. . E, EN, ES FORMAT - E 10. , E 0.xxxxE yy 0.xxxxE yyy 0.xxxx yyy ( - yyy 99).

WRITE (*, "(E10.2)") 123D5

0.12E+08

ES E E 1 10.

WRITE (*, "(ES10.2)") 123D5

1.23E+07

EN E 3 E 1 1000.

WRITE (*, "(EN10.2)") 123D5

12.30E+06

9.3729

READ:

READ (*, "(F8.2)") x ! E8.2 EN8.2 ES8.2

93.729E-1

FORMAT 87

READ 10

READ (*, "(F10.1)") x ! E10.1 EN10.1 ES10.1

, - F (), 0 .

format , . ..

WRITE (*,"(F6.1,E8.1)") (0.1d0, 100.0d0)

0.10.1E+03

, format ( - ) ( ) : ..

WRITE (*,"(F6.1,'+i',E8.1)") (0.1d0, 100.0d0)

0.1+i0.1E+03

L - , w. .TRUE. .FALSE. - T F . .

A, . - , . ..

WRITE (*,"(A)") "Thisisastatement"

Thisisastatement

G , w, , d, Gw.d. , Iw, Lw, Aw Ew.d F.

88 /

( 6), , , format , TYPE.

, format . format . format , FORMAT . format , , . ,

100 FORMAT (I4,I4,I4)

100 FORMAT (3I4)

200 FORMAT (I4,F4.2,I4,F4.2)

200 FORMAT (2(I4,F4.2))

format . * FORMAT .

7.2.1 FORMAT

, FORMAT ( )

, ,

/, ,

nX, n . format n .

, + - , FORMAT sp, format . ss FORMAT . ,

89

WRITE (*,"(F8.2,sp,F8.2)") 5.12d0, 2.3d0WRITE (*,"(F8.2)") 9.16d0WRITE (*,"(F8.2,sp,F8.2,F5.1,ss,F5.1)") 5.12d0, 2.3d0, 4.1d0, 3.5d0

5.12+2.309.165.12+2.30+4.13.5

7.2.2

, WRITE advance="no". FORMAT ( *). ..

WRITE (*, "(A)", advance="no") "Givenumber:"READ (*,*) x

. -.

7.3

, . , , .

.

7.3.1

. . OPEN .

OPEN(unit = u, file = fname, status = stat, position = pos, &action = act)

unit= . u , , - . 99. , file= .

90 /

, . , - input.dat 12,

OPEN(unit = 12, file = "input.dat")

OPEN(12, file = "input.dat")

. , , .

:

status=stat, stat old, new, replace, scratch,unknown.

1. ( ),2. ( ),3. , -

,4. (

file=),5. .

, - status="unknown". status="old" status="new" , .

position=pos, pos

1. rewind, ,

2. append, , 3. asis ,

OPEN. - position=.

action=act act read, write readwrite , , .

.. 34, data, ,

91

OPEN (unit = 34, file ="data", status = "old", action = "read") READ WRITE ,

*, READ (unit = 34, fmt = "(F8.2)") x, ,READ (34, "(F8.2)") x

. CLOSE (unit = u), CLOSE (u) u . , .

READ , . READ . , ( ).

-

(filesystem) - (path), .

. 5 6 . 0 ( ). - .

. .

7.3.2

- , 2.6.1, - , . - .

92 /

, . READ/WRITE, *. , ,

CHARACTER(50) :: strINTEGER :: i

i = 3WRITE (str, "(A,I0,A)") "fname_", i, ".dat" ! str "fname_3.dat"

FORMAT str. OPEN:

OPEN(unit = 43, file = str)

, 43 fname_3.dat.

.

93

7.4

1. - trig.dat. - , 0 359.9 0.1. 4 : ,, . .

- trig.dat .

2. calendar ( 2014):

03/2014

1 23 4 5 6 7 8 910 11 12 13 14 15 1617 18 19 20 21 22 2324 25 26 27 28 29 3031

:

() (, , ) .() .

, Zeller 9 39. , 8 39.

3. 1.0 9.9 0.1. 3 , :

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91 0.000 0.041 0.079 0.114 0.146 0.176 0.204 0.230 0.255 0.2792 0.301 0.322 0.342 0.362 0.380 0.398 0.415 0.431 0.447 0.462...

......

......

......

......

......

9 0.954 0.959 0.964 0.968 0.973 0.978 0.982 0.987 0.991 0.996

. i j , (i, j).

94 /

4. Pascal. ,n n,

P (i, 1) = P (1, j) = 1 i, j P (i, j) = P (i 1, j) + P (i, j 1) i, j > 1.

.

Pascal. . n = 7.

pascal.

5. ( -):

2. 3 -

1.

n, . Collatz, , , 1. n.

collatz.dat 2 100000 ( ) 1 ( ).

, .

6. () random.txt 10000 - [20, 20].

() random.txt , 0 .

7. Lemoine, 5, (

95

)1. m 7 999999: p, q m = p + 2q. m, p, q, , lemoine.dat, .

8. - :

: P1. : (

). 0 1 pixels

: pixel 0 1. . 70.

plain pbm (portable bitmap) .

512 512 , - (, ), 1 0. , . .ppbm plain pbm.

9. (ASCII) :

: P2. : (

). K

. 256. 255.

pixels . pixel - 0 K . . 70 . pixel .

1 1 .

96 /

plain pgm (portable graymap) . input.ppgm - output.ppgm : pixel (i, j) pixel (i, j) ( p). , (i, j) , pixels - (i p, j p). (i, j) , . p. o , http://fla.st/1KxdatB.

10. - :

: P3. :

( ). K, 255,

. 255.

pixels . - (R), (G), (B) pixel, . pixel K 0 0 ( K 255 255 0 0). pixel 0 K 0. 0 0 0 K K K . K K 0. RGB 0 K . 70 .

plain ppm (portable pixmap) . france.pppm 2, 512 768 pixels, .

11. Game of Life3 Dr. J. Conway. - .

2 : , , .3http://www.math.com/students/wonders/life/life.html

http://fla.st/1KxdatBhttp://fla.st/1KxdatBhttp://www.math.com/students/wonders/life/life.html

97

MN , ( ). K . ( ):

() - ().

() ().

() ( !).

- . plain pbm ( 8), 4. 512 512 1000 . - , (, ).

12. (Langtons ant5) 128 128. . . ( (64, 64)), :

, , 90 .

, , 90 .

12000. 100 plain pbm ( 8). ;

13. 12 2 6,3 4, 2 2 3.

4 UNIX, imagemagick, animate*.ppbm

5http://mathworld.wolfram.com/LangtonsAnt.html

http://mathworld.wolfram.com/LangtonsAnt.html

98 /

. , ( 1 ). . , . , 12 : 2 2 3, , ..13, : 13.

8

8.1

- Fortran, - . , , , , , , . , , . , , , , , . , . , - .

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

99

100

(structured) (procedural) -.

Fortran , , , PROGRAM END PROGRAM . , . -.

8.2

- . -

f(x) = x2 + 5x 2 .

, x2 + 5x 2, , f , - . , x, , . f(x) - f , , x. x . x2+5x2 x ( ) f , x. , ..,y = f(2.5) y = 2.52+52.52. - x, , , x. () . (). , . , .

Fortran, : - . , . . , - , . .

101

( ). - , , - . , . - - , 6. , , .

_ FUNCTION (, ,)__ :: __ ::

:: _, ... :: _, ...

...... ! END FUNCTION

, (, , ):

FUNCTION (, ,)__ :: __ :: _ ::

:: _, ... :: _, ...

...... ! END FUNCTION

, , .

( 6).

102

. , , , , , ( ). .

SUBROUTINE (, ,)__ :: __ ::

:: , ...

! ......END SUBROUTINE

- .

8.3

:

x2+x9 x = 1.3, x = 2.7, x = 0.5. :

PROGRAM funIMPLICIT NONE

DOUBLE PRECISION :: x1, x2, x3, y1, y2, y3

x1 = 1.3d0y1 = x1**2 + x1 - 9.0d0

x2 = 2.7d0y2 = x2**2 + x2 - 9.0d0

x3 = -0.5d0y3 = x3**2 + x3 - 9.0d0

! ......END PROGRAM fun

103

, , - . , - .

FUNCTION polynomial(x)IMPLICIT NONE

DOUBLE PRECISION, INTENT (in) :: xDOUBLE PRECISION :: polynomial

polynomial = x*x + x - 9.0d0END FUNCTION polynomial

( polynomial) - , 2.1.2. - , , x, . . , , - . , , -, ,

DOUBLE PRECISION FUNCTION polynomial(x)..........END FUNCTION polynomial

-, , . . -, .

. , , , x . , INTENT (in). . - - . , ,

104

( ) INTENT (out). , INTENT (inout). , . .

( ), . , .

, , :

ln(1 + x) = k=1

(x)k

k, 1 < x 1 .

ln(1.05) - (LOG()). , .

PROGRAM logarithmIMPLICIT NONEDOUBLE PRECISION :: sum, y, x, termINTEGER :: k

! PART A: READDO

READ *, yx = y - 1.0d0IF ( (-1.0d0 < x) .AND. ( x

105

END DO

! PART C: PRINTWRITE (*,*) "ln(", y, ")=", sumWRITE (*,*) "Timiapoenswmatomenisynartisi", LOG(y)

END PROGRAM logarithm

:

1. ,

2.

3. .

-. :

sum ( ) !PART B !PART C. - sum:

FUNCTION sumseries(x)IMPLICIT NONE

DOUBLE PRECISION, INTENT (in) :: xDOUBLE PRECISION :: sumseries

DOUBLE PRECISION :: term, sumINTEGER :: k

k = 1sum = 0.0d0DO

term = -(-x)**k / kIF (ABS(term) < 1d-6) EXITsum = sum + termk = k + 1

END DOsumseries = sum

END FUNCTION sumseries

- . , .

106

, -, . . ( , , ) .

FUNCTION readxy(y)IMPLICIT NONEDOUBLE PRECISION, INTENT (out) :: yDOUBLE PRECISION :: readxy

DOUBLE PRECISION :: x

DOREAD *, yx = y - 1.0d0IF ( (-1.0d0 < x) .AND. ( x

107

8.3.1

, , , , , , , , , . : , interface () . interface , - . , ( ), ( END). .. interface polynomial() :

INTERFACEFUNCTION polynomial(x)

IMPLICIT NONEDOUBLE PRECISION, INTENT (in) :: xDOUBLE PRECISION :: polynomial

END FUNCTION polynomialEND INTERFACE

readxy() :

INTERFACESUBROUTINE readxy(x,y)

IMPLICIT NONEDOUBLE PRECISION, INTENT (out) :: x, y

END SUBROUTINE readxyEND INTERFACE

INTERFACE .

8.4

, , (, , 1).

1 .

108

- , , 2. , , , , . , .. polynomial() :y = polynomial(1.3d0)

y = polynomial(2.4d0) + 2.0d0 * polynomial(3.5d0) + 4.0d0

, ,WRITE (*,*) polynomial(4.6d0)

, , , 3:polynomial(3.4d0) ! polynomial(3.4d0) = 4.0 ! y = polynomial !

. , , . , - CALL . readxy() (!)DOUBLE PRECISION :: x, y

CALL readxy(x,y)

. ( INTENT (out)), . , x, y .

8.5

(logarithm), 8.3 104. , , sumseries() readxy()., :

2 , 8.7.3. keywords .

3 .

109

FUNCTION sumseries(x)IMPLICIT NONE

DOUBLE PRECISION, INTENT (in) :: xDOUBLE PRECISION :: sumseries

DOUBLE PRECISION :: term, sumINTEGER :: k

k = 1sum = 0.0d0DO

term = -(-x)**k / kIF (ABS(term) < 1d-6) EXITsum = sum + termk = k + 1

END DOsumseries = sum

END FUNCTION sumseries

SUBROUTINE readxy(x,y)IMPLICIT NONEDOUBLE PRECISION, INTENT (out) :: x, y

DOREAD *, yx = y - 1.0d0IF ( (-1.0d0 < x) .AND. ( x

110

SUBROUTINE readxy(x,y)IMPLICIT NONEDOUBLE PRECISION, INTENT (out) :: x, y

END SUBROUTINE readxyEND INTERFACE

CALL readxy(x,y) ! READ

sum = sumseries(x) ! COMPUTE

WRITE (*,*) "ln(", y, ")=", sum ! PRINTWRITE (*,*) "Timiapoenswmatomenisynartisi", LOG(y)

END PROGRAM logarithm

. , - , .

8.6 RETURN

, , . , END FUNCTION /END SUBROUTINE RETURN . , .

RETURN : - . , , - .

SUBROUTINE riza(x)IMPLICIT NONEDOUBLE PRECISION, INTENT (in) :: x

IF (x < 0.0d0) THENWRITE (*,*) "Wrongargument"RETURN

END IF

WRITE (*,*) "Therootis", SQRT(x)END SUBROUTINE riza

111

, 4 .

8.7

, , -. , . , , 5.

:

8.7.1

. :

, ( -) . ..

SUBROUTINE athr(a)IMPLICIT NONEDOUBLE PRECISION, INTENT (in) :: a(3)

WRITE (*,*) a(1) + a(2) + a(3)END SUBROUTINE athr

3 .

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

SUBROUTINE def(a)IMPLICIT NONEDOUBLE PRECISION, INTENT (in) :: a(:,:)

4 , 8.8.5 , 8.7.3, keywords

.

112

WRITE (*,*) SUM(a)END SUBROUTINE def

. - SIZE(). , ( )

DOUBLE PRECISION :: b(SIZE(a,1), SIZE(a,2))

( ), ., a(m,n)

SUBROUTINE def(a, m, n)IMPLICIT NONEINTEGER, INTENT (in) :: m,nDOUBLE PRECISION, INTENT (in) :: a(m,n)

WRITE (*,*) SUM(a)END SUBROUTINE def

, .

INTENT .

ALLOCATABLE 6., - (ALLOCATE/DEALLOCATE ). - ( ) ALLOCATABLE .

8.7.2

- . : - , INTERFACE .

6ISO TR 15581 Fortran 2003.

113

, , , . - . ( ) , . :

SUBROUTINE plot(a, b, f)IMPLICIT NONEDOUBLE PRECISION, INTENT (in) :: a, b

INTERFACEFUNCTION f(x)

IMPLICIT NONEDOUBLE PRECISION, INTENT (in) :: xDOUBLE PRECISION :: f

END FUNCTION fEND INTERFACE

INTEGER :: iDOUBLE PRECISION :: x, step

step = (b-a) / 100.0d0

DO i=0,100x = a + i * stepWRITE (*,*) x, f(x)

END DOEND SUBROUTINE plot

FUNCTION g(x)IMPLICIT NONEDOUBLE PRECISION, INTENT (in) :: xDOUBLE PRECISION :: g

g = x*x - 2.0d0END FUNCTION g

FUNCTION h(x)IMPLICIT NONEDOUBLE PRECISION, INTENT (in) :: xDOUBLE PRECISION :: h

h = 1.0d0 - 2.0d0 * x

114

END FUNCTION h

PROGRAM testIMPLICIT NONEINTERFACE

SUBROUTINE plot(a, b, f)IMPLICIT NONEDOUBLE PRECISION, INTENT (in) :: a, b

INTERFACEFUNCTION f(x)

IMPLICIT NONEDOUBLE PRECISION, INTENT (in) :: xDOUBLE PRECISION :: f

END FUNCTION fEND INTERFACE

END SUBROUTINE plot

FUNCTION g(x)IMPLICIT NONEDOUBLE PRECISION, INTENT (in) :: xDOUBLE PRECISION :: g

END FUNCTION g

FUNCTION h(x)IMPLICIT NONEDOUBLE PRECISION, INTENT (in) :: xDOUBLE PRECISION :: h

END FUNCTION hEND INTERFACE

CALL plot(0.0d0, 1.0d0, g)CALL plot(0.0d0, 1.0d0, h)

END PROGRAM test

plot , f, 101 a, b . g(x), h(x) 0.0, 1.0.

- , , . Fortran 66

115

.

8.7.3

Fortran , OPTIONAL . . , . , .

PRESENT() Fortran 95 , - , . , .. a, , PRESENT(a) .TRUE.., PRESENT(a) .FALSE. .

:

FUNCTION f(x,a)IMPLICIT NONEDOUBLE PRECISION, INTENT(in) :: xDOUBLE PRECISION, INTENT (in), OPTIONAL :: aDOUBLE PRECISION :: f

IF (PRESENT(a)) THENf = x + a

ELSEf = x + 10.0d0

END IF

END FUNCTION f

, f() . , -, .

f() ,

PROGRAM addIMPLICIT NONEINTERFACE

FUNCTION f(x,a)IMPLICIT NONEDOUBLE PRECISION, INTENT(in) :: x

116

DOUBLE PRECISION, INTENT (in), OPTIONAL :: aDOUBLE PRECISION :: f

END FUNCTION fEND INTERFACE

WRITE (*,*) f(3.0d0)WRITE (*,*) f(3.0d0, 20d0)

END PROGRAM add

8.8

- . .

- ( , , ). - SAVE , , .

SUBROUTINE f(a)IMPLICIT NONEDOUBLE PRECISION, INTENT (in) :: a

INTEGER, SAVE :: k = 0

k = k + 1

! ...END SUBROUTINE f

k . f().

SAVE . . , .

(recursive) 117

8.9 (recursive)

. ..

n! =

{1 2 (n 1) n = (n 1)! n , n > 0 ,1 , n = 0 .

n , ( n1). , , ,, . , , .

Fortran 95 . , :

,

. .

(recursive) Fortran 95.

FUNCTION par(n) ! IMPLICIT NONEINTEGER, INTENT (in) :: nINTEGER :: par

IF (n == 0) THENpar = 1

ELSEpar = par(n-1) * n

END IFEND FUNCTION par

118

- ( n ). - Fortran, . - par() par(n)par(n-1) par(n-2) . . . ( - ) 0.

, Fortran 95 ( - ).

RECURSIVE FUNCTION par(n) RESULT (p)IMPLICIT NONEINTEGER, INTENT (in) :: nINTEGER :: p

IF (n == 0) THENp = 1

ELSEp = par(n-1) * n

END IFEND FUNCTION par

RECURSIVE () ( RESULT) ( ) . , , . .

( - , RECURSIVE ), Fortran 95 - Fortran 77.

:

. , Hermite, - .

H0(x) = 1 ,

H1(x) = 2x ,

(ELEMENTAL) 119

H2(x) = 4x2 2 ,

H3(x) = 8x3 12x ,

H4(x) = 16x4 48x2 + 12 ,

...

Hn(x) :

Hn(x) = 2xHn1(x) 2(n 1)Hn2(x) , n 2 .

(H0(x) = 1) (H1(x) = 2x) . , H1(x) H2(x) H3(x), .

- Hermite, Hn(x), x, :

RECURSIVE FUNCTION hermite(n,x) RESULT (h)IMPLICIT NONE

INTEGER, INTENT (in) :: nDOUBLE PRECISION, INTENT (in) :: xDOUBLE PRECISION :: h

IF (n==0) h = 1.0d0IF (n==1) h = 2.0d0 * xIF (n>=2) h = 2.0d0 * (x * hermite(n-1,x) - (n-1) * hermite(n-2,x))

END FUNCTION hermite

: ( , -, ) RECURSIVE RESULT .

8.10 (ELEMENTAL)

, . .. , . , Fortran . ,

DOUBLE PRECISION :: x, y

120

x = 5.3d1y = SQRT(x)

DOUBLE PRECISION, DIMENSION (100) :: x, yx(1) = ....x(2) = .......x(100) = ....

y = SQRT(x)

y(1) = SQRT(x(1))y(2) = SQRT(x(2))...y(100) = SQRT(x(100))

ELEMENTAL . .. ELEMENTAL FUNCTION myexp(x)

IMPLICIT NONEDOUBLE PRECISION, INTENT (in) :: xDOUBLE PRECISION :: myexp

INTEGER :: iDOUBLE PRECISION :: term

myexp = 0.0d0term = 1.0d0DO i = 0, 10

myexp = myexp + termterm = term * x / (i+1)

END DOEND FUNCTION myexp

( ), . - .

ELEMENTAL - (8.8), STOP (2.10.1 . , -.

MODULE 121

ELEMENTAL - INTENT (in). ELEMENTAL INTENT, INTENT (in).

8.11 MODULE

, ( ), -, . , . .

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

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

MODULE onoma

END MODULE onoma

MODULE , 2.1.2. - , - ( 6), (INTERFACE ), ( CONTAINS), .

MODULE MODULE,

USE onoma

122

( IMPLICIT NONE). , PROGRAM ... FUNCTION ... SUBROUTINE ... MODULE .... MODULE, MODULE7.

:

MODULE constantsIMPLICIT NONE

DOUBLE PRECISION, PARAMETER :: pi = 3.14159265358979323846d0END MODULE constants

PROGRAM mainprogUSE constants

IMPLICIT NONE!.....

! use piEND PROGRAM mainprog

FUNCTION f(x)USE constants

IMPLICIT NONE!.....

! use piEND FUNCTION f

, MODULE . , , . module , - .

, - : MODULE . INTERFACE , INTENT, . -, (

7 PRIVATE .

MODULE 123

) MODULE . - . , modules , . .

module Fortran 90 - (object based) .

124

8.12

1. x ex2/2. x = 0.3, 1.4, 5.6.

2. .

3. () . 3!, 5!, 7!.

4. - . - (3.2, 5.6, 8.1).

5. , , ( ;). 89, 261, 1511.

6. .

7. . (,, MINVAL().)

8. .(, , MAXLOC().)

9. . , .

10. ,

i aibi a, b . (, -

DOT_PRODUCT ().)

11. [a, b] .

12. , , http://tinyurl.com/114rndint.

http://tinyurl.com/114rndint

125

-. 5744, 6789, 2774.

13. , - , Mersenne. k Mersenne k+1 2. Mersenne 10000.

14. ex

ex x0/0! + x1/1! + x2/2! + + x12/12! .

ex x = 0.5, 1.2, 4.1.

15.

sinx x1/1! x3/3! + x5/5! x7/7! + x9/9! x11/11! .

- . 35.

16.

() ex

ex =n=0

xn

n!,

() sinx

sinx =k=0

(1)kx2k+1

(2k + 1)!,

() cosx

cosx =k=0

(1)kx2k

(2k)!.

- . - 1010.

126

17. . +, -, *, /. - - ( RETURN). , - . 2 + 3i,5.7 9i.

18. - . - (C) (F ) . F = 9/5C + 32. - :

22 C, 0 (273.15 C), (6000 C).

100F.

19. ax2 +bx+ c = 0 . a, b, c, . - .

20. - http://tinyurl.com/114matrixA http://tinyurl.com/114matrixB. - . , , (1, 1), (1, 2), , (2, 1), (2, 2), .

. . ,.

( ).

, - , , 4 .

http://tinyurl.com/114matrixAhttp://tinyurl.com/114matrixB

127

, .

21. - . .

2.3 i 1 7i 5.8 2.9 3.7i4.9i i 9 0.3i 2 + 0.72i8.2 + 4i 0.8 + i 0.2 + 5i 9 3i2.3i 7.1 + 9i 0.9 4i

;

22. - x(t) = A cos(t)+B sin(t), A =3 cm, B =2 cm, =12Hz.

() t x.

() data t x(t) 4 , t = 0.0, 0.5, 1.0, ,100.0 s. .

23. ()

n! =

{(n 1)! n , n > 0 ,1 , n = 0 .

24. , , 2. . 4096, 65534, 1855932 2.

25. () , , 10 .

() - . - ( ) , . .. 19791, 4774 . .

128

() palindrome.dat . .

26. palindrome.dat - ( 25) 1000000. .

27. digit ,N d. d N. N . . , digit N=57960,d=2 6. d N, 0. http://tinyurl.com/ints201411. 3590 , . . , digit.txt, .

28. ( ). {2, 8, 8, 3, 5, 5, 5, 8, 8, 1, 6, 7, 7, 7} 7.

29. r = (x, y, z) x, r

Rx() =

1 0 00 cos sin 0 sin cos

,, xy

z

=1 0 00 cos sin 0 sin cos

xy

z

. y, z

Ry() =

cos 0 sin

0 1 0sin 0 cos

http://tinyurl.com/ints201411http://tinyurl.com/ints201411

129

Rz() =

cos sin 0sin cos 0

0 0 1

.()

. , .

() .

() :i. x = 0.5, y = 0.3, z =

1.2. 30 y, 35 x 58 z. .

ii. ( ) .

30. Hermite,Hn(x). n , 0, 1,. - Hermite

H0(x) = 1

H1(x) = 2x

H2(x) = 4x2 2

... =...

Hermite

Hn(x) = 2xHn1(x) 2(n 1)Hn2(x) , n 2 .

Hermite. n x . Hn(x).

31. ( m V = kx2/2)

n(y) =

1

2nn!

(km

h

)1/4Hn(y)e

y2/2 , (8.1)

130

y = x

km/h. Hermite 30 () - (8.1). , n,x. m = k = h = 1. harmonic.dat - n = 5 60 x [6, 6], x ( , x ).

32. Legendre,P(x), x [1, 1]. , 0, 1,. Legendre P0(x) = 1 P1(x) = x, :

P(x) = (2 1)xP1(x) ( 1)P2(x) .

Legen-dre. - x . P(x).

33. , -, . http://tinyurl.com/114rndint , . . .

34. N 1 x < 1, 1 y < 1. 1 (x2 + y2 = 1) . N (, ). .. N = 103,N = 104, , N = 109. N 2 2 ( ;)

35. , (x, y) (, ) . plotdata x, y , =0,2,4,,360 = esin 2 cos(4) + sin5((2 )/24). () . .

http://tinyurl.com/114rndint

131

36. N . 1000.

37. Cliff Pickover8:

xi+1 = |(100 ln(xi)) mod 1| ,

x0 = 0.1. a mod 1 a. - xi [0, 1). 9 ( INTENT (out) ).

38. - A N N 10

detA =Ni=1

(1)i+jaij det Aij ,

j, .. 1. A i j aij , Aij A i j .

39. 38 - Cramer11.

40. ( )

nm(r, , ) = Rn(r)Ym(, ) .

,

Ym(, ) =

2+ 1

4

(m)!(+m)!

Pm (cos )eim .

Legendre, Pm (x),

= m

Pm (x) = (1)m 1 3 5 (2m 1) (1 x2)m/2 ,

8http://mathworld.wolfram.com/CliffRandomNumberGenerator.html9 8.8.10http://mathworld.wolfram.com/DeterminantExpansionbyMinors.html11http://mathworld.wolfram.com/CramersRule.html

http://mathworld.wolfram.com/CliffRandomNumberGenerator.htmlhttp://mathworld.wolfram.com/DeterminantExpansionbyMinors.htmlhttp://mathworld.wolfram.com/CramersRule.html

132

= m+ 1Pm (x) = x(2m+ 1)P

mm (x) ,

(m)Pm (x) = x(2 1)Pm1(x) (l +m 1)Pm2(x) .

[0, ] [0, 2) .

-.

Legendre,Pm (x).

, Ym(, ). 50100

Ym(, ). ylm_data sin cos, sin sin,cos , Ym(, )Y m(, ) (, x, y, z, ) , = 2, m = 0 (, d-).

41. f(x), , x f(x) (f(x) = 0), Mller.

() x0, x1, x2 - .

()

q =x2 x1x1 x0

,

A = q (f(x2) f(x1)) q2 (f(x1) f(x0)) ,B = (q + 1) (f(x2) f(x1)) +A ,C = (q + 1)f(x2) .

()

x3 = x2 2C(x2 x1)

D,

D B +

B2 4AC, B

B2 4AC .

() 41.() x0 x1, x1 x2, x2 x3.

41.

133

() .

, xn, q, A, B, C, D . f(x) = x3 x+1 - Mller.

42. Bessel , n, Jn(x), -

Jn(x) =

m=0

(1)m

m!(m+ n)!

(x

2

)2m+n.

bessel.dat J0(x), J1(x),J2(x) 150 xi [0, 20].

xi J0(xi) J1(xi) J2(xi)

I: , , . 1012. II: . .

43. a - : , x0. x0 = 1.

xi+1 = xix3i + 2a

2x3i + a

x1, x2, . . .. ,

x1 = x0x30 + 2a

2x30 + a,

x2 = x1x31 + 2a

2x31 + a, .

x1, x2, . . . 3a.

xk |x3k a| , , .. 1012. .

134

20.0, 20.1, 20.2,,30.0. cbrt : 20.0, 20.1, 20.2,,30.0 . 12 .

44. . , 3 3 , , N N , (.. X O) (, , N-) , , . , . - . . , , , , , .

X|O|X| |O

O|X|

, , ( ) - .

45. four-in-a-row . , M N ( 7 6 ). . , . , . . - . , ( ). , .

46. N . NN , N > 3, N , . .

135

. - - .

:

N N . - board. 0 .

N ..column. . i column(i).

, (, , ) board. , - ( 0) ( 1). -.

. 1 , . .

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

, (N N , Q _).

47. Sudoku. sudoku. - 9 9 , 33 19, ( ).

136

, . :

() 1.

() -. , 2, 3, . - . , .

() -. 19, . , - . , , , ...

5 3 76 1 9 5

9 8 68 6 34 8 3 17 2 6

6 2 84 1 9 5

8 7 9

48. fractal : , (x = 0, y = 0),

137

(x, y)

x = a x+ b y + ey = c x+ d y + f

a,b,c,d,e,f . - fractal (, x x y y (x, y)). . (x, y) , fractal. :

() in.dat 4 . - 7 : 6 a, b, c, d, e, f p . . ,

pi, 1.

() r [0, 1). . , 0 r < p1 , p1 r < p1 + p2 .

() 1000 fractal.dat.

in.dat

0 0 0 0.16 0 0 0.010.85 0.04 0.04 0.85 0 1.6 0.850.2 0.26 0.23 0.22 0 1.6 0.070.15 0.28 0.26 0.24 0 0.44 0.07

0 0 0 0.25 0 0.4 0.020.95 0.005 0.005 0.93 0.002 0.5 0.840.035 0.2 0.16 0.04 0.09 0.02 0.070.04 0.2 0.16 0.04 0.083 0.12 0.07

, fractal.dat .

49. - . . 10 10 - . , , :

138

() 1 (5 ),() 1 (4 ),() 1 (3 ),() 1 (3 ),() 1 (2 ).

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

50. p(z). z0 ( p(z0) = 0) p(z), p(z), :

zi+1 = zi p(zi)

p(zi), i = 0, 1, 2, . . . .

, z0, z1. z2, ., p(z), |p(zi)| . - NewtonRaphson , . p(z) = z3 1 a =1, b = ei2/3, c = ei2/3. , z = 0 + i0, . . , .., a (RGB = (255, 0, 0)). b (RGB = (0, 255, 0)) c (RGB = (0, 0, 255)). 0 + i0 (RGB = (255, 255, 255)).

() N = 512 [1, 1] ( ): xi, i = 0, . . . , N 1.

139

() M = 512 [1, 1] ( ): yj , j = 0, . . . ,M 1.

() z = xi + iyj .

() pixels (i, j) newton.pppm. plain ppm ( 10 96). newton.pppm Newtonfractal.

140

.1

.1.1

. (1 +N)/2 , N .

.1.2

N , 1+log2N , ( N ) , ., .

, , :

( ) :

, .

, , , .

, , , .

141

142

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

- :

0, .

1, . , .

1, ( ) . , , . , .

.1.3 hash

, (hash table), . - : . , , .

-, , (hash function). - (hash). hash . , ..

- (), .

16 bit, [0, 655