lit
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DERIVATIVE1 -

D400

: H.von Eicken, .. :

popaa c a po pooo f(x) o x=x1. poaac, o f(x) ppa opcoc o x1.

:

: - FUNCTION
: - DERIVATIVE1
: - F - /-
COMMON BLOCK: - /D400_ERROR/ NERROR

:

DF = DERIVATIVE1(F,X1)
:

X1 - (REAL*8) ap x1;
F - (REAL*8) opopa- o ooo apa,
c f(x).
F o oca a popa opaopo EXTERNAL.

:

DF - (REAL*8) x=x1;
/D400_ERROR/NERROR - (INTEGER) .
:
 0 - ;
 1 - ,
     ;
-1 - , .

:

co o [1] coo ppoa c pacpocpa a co ppoa, [2]. cc pa pooa f(x), pp ac (a,b), x=x1, a<=x1<=b. Bcc a aaa T-ca:

     0
    T
     0       0
            T
     1       1       0
    T               T
     0       1       2       0
            T               T               0
     2       1       1       3      .      T
    T               T       .       .       m
     0       2       2      .
            T       .       .
     3       1      .
    T       .       .
     0      .
    .       .
    .
    .
     k
    T
     0

Tk = (f(x1+ΔXk)-f(x1-ΔXk))/2*ΔXk,
ΔXk = DELTA*LAMBDAk.
DELTA - aa a, oopa op pcoc x1.
LAMBDAk = 1, 3/4, 1/2, 3/8, 1/4, ... .

 
               /   m  k+1          k
               |  2 *T    - 1.125*T
               |      m-1          m-1            m - o,
               | ---------------------            k - o;
        m      |       m
      0 |      2  - 1.125
               |
               |          m   k+1    k
               | 1.125 * 2 * T    - T
       k       |              m-1    m-1          m - o,
      T   =   <  -----------------------          k - o;
       m       |    m
               |   2  * 1.125 - 1
               |
               |  m    k+1    k
               | 2  * T    - T
               |       m-1    m-1
               | ----------------                 m - o.
               |       m
               \      2 -1

Bco a Tm0 o a popa.
a coa a, popaa paoa c T-co, ac oopo 7 (k=0,1,...,7), a o caa aa a LAMBDAk 0.125.
Ha po a popaa c ac T-c, .. T00 - T0k.
c cp oc, opo a.
poo ca poaa coa pooa o pop c DELTA=DELTA*0.1, oa DELTA <= 10-5.
c oc c ao , poco opa a popa NERROR=1, DERIVATIVE1=0.

Ha opo a popaa c oc co c pop a co a oc.
c oc poco, o popc, oo cao ao a DELTA. c oco, o aac pac c o DELTA. NERROR=-1, DERIVATIVE1.

:

  1. Romberg W. Vereinfachte numerische Integration.
    Det. Kong. Norske Viedenskabers Selskab Forhandlinger, 28, Nr.7, Trondheim, 1955.
  2. Rutishauser H. Ausdehnung des Rombergschen Prinzips,
    Numerishe Mathematik, 5, 48-54, 1963.
:
       . . .
       IMPLICIT REAL*8 (A-H,O-Z)
       EXTERNAL F
       X1=-0.5D0
       DF=DERIVATIVE1(F,X1)
       WRITE(*,*) X1,DF
       . . .
       DOUBLE PRECISION FUNCTION F(X)
       IMPLICIT REAL*8 (A-H,O-Z)
       F=DCOS(X)/DSIN(X)
       RETURN
       END
:
       X1= -5.000000000000000E-001    DF= -4.350685299341246 


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