# chapter 5 6

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```					5.6 Multistep Methods
The methods discussed to this point in the chapter are called one-step methods because the

approximation for the mesh point ti 1 involves information from only one of the previous mesh

points, ti . Although these methods might use functional evaluation information at points between

ti and ti 1 , they do not retain that information for direct use in future approximation. All the

information used by these methods is obtained within the subinterval over which the solution is
being approximated.

Since the approximate solution is available at each of the mesh points t0 , t1 ,                      , ti before the

approximation at ti 1 is obtained, and because the error w j  y t j                  tends to increase with j ,

it seems reasonable to develop methods that use these more accurate previous data when

approximating the solution at ti 1 .

Method using the approximation at more than one previous mesh point to determine the
approximation at the next point are called multistep method. The precise definition of these
methods follow, together with the definition of the two types of multistep methods.

Definition 5.14
An m-step multistep method for solving the initial-value problem

y '  f t, y  , a  t  b, y  a    ,                                     (5.22)

has a difference equation for finding the approximation wi 1 at the mesh point ti 1 represented

by the following equation, where m is an integer greater than 1:
wi 1  am1wi  am2 wi 1              a0 wi 1 m
 h bm f  ti 1 , wi 1   bm1 f  ti , wi 
                                                                    (5.23)

      b0 f  ti 1 m , wi 1 m   ,

for i  m  1, m,    , N  1 , where h   b  a  N , the a0 , a1 ,                 , am1 and b0 , b1 ,   , bm are

constants, and the starting values

w0   , w1  1 , w2   2 ,                , wm1   m1

are specified.                                                                                               ■

When bm  0 the method is called explicit, or open, since Eq. (5.23) then gives wi 1 explicitly

in terms of previously determined values. When bm  0 the method is called implicit, or closed,

since wi 1 occurs on both sides of Eq. (5.23) and is specified only implicitly.
EXAMLE 1
The equations

w0   , w1  1 , w2   2 , w3   3 ,
h                                                                                                    (5.24)
wi 1  wi       55 f  ti , wi   59 f  ti 1 , wi 1   37 f  ti  2 , wi  2   9 f  ti 3 , wi 3   ,
                                                                                             
24

for each i  3, 4,       , N  1 , define an explicit four-step method known as the fourth-order

w0   , w1  1 , w2   2 ,
h                                                                                                (5.25)
wi 1  wi        9 f  ti 1 , wi 1   19 f  ti , wi   5 f  ti 1 , wi 1   f  ti  2 , wi  2   ,
24                                                                                          

for each i  2,3,        , N  1 , define an implicit three-step method known as the fourth-order

The staring values in either (5.24) or (5.25) must be specified, generally by assuming w0   and

generating the remaining values by either a Runge-Kutta method or some other one-step
technique.

To apply an implicit method such as (5.25) directly, we must solve the implicit equation for wi 1 .

It is not clear that this can be done in general or that a unique solution for wi 1 will always be

obtained.
To begin the derivation of a multistep method, note that the solution to the initial-value problem

(5.22), if integrated over the interval        ti , ti1  , has the property that
y  ti 1   y  ti             y '  t  dt               f  t , y  t   dt.
ti1                         ti1

ti                             ti

Consequently,

y  ti 1   y  ti                 f  t , y  t   dt.
ti1
(5.26)
ti


Since we cannot integrate f t , y  t             without knowing y  t  , the solution to the problem, we

instead integrate an interpolating polynomial P  t  to f t , y  t                                that is determined by some of

the previously obtained data points          t0 , w0  , t1, w1  ,              , ti , wi  . When we assume, in addition,

that y  ti   wi , Eq. (5.26) becomes

y  ti 1   wi   P  t  dt.
ti1
(5.27)
ti

Although any form of the interpolating polynomial can be used for the derivation, it is most
convenient to use the Newton backward-difference formula.
To derive an Adams-Bashforth explicit m-step technique, we form the backward-difference
polynomial

Pm1  t  through                    t , f t , y t  , t
i         i           i                  i 1   , f  ti 1 , y  ti 1   ,                                          
, ti 1m , f  ti 1m , y ti 1m   .

Since P 1  t  is an interpolatory polynomial of degree m 1 , some number i in
m                                                                                                                                      ti1m , ti  exists
with

f  t , y  t    Pm1  t  
f
m
 , y  
i        i
t  ti t  ti1  t  ti1m  .
m!

Introducing the variable substitution t  ti  sh , with dt  h ds into P 1  t  and the error
m

term implies that

ti 1 m 1                        s  k
f  t , y  t   dt                 1                    f  ti , y  ti  dt
ti 1

k
ti                                    ti
k 0                    k
f
m
 , y    
 t  ti  t  ti 1   t  ti 1m  dt
ti 1

i         i
ti                      m!
m 1
 s 
   k f  ti , y  ti   h  1
1
0  k  ds
k

k 0                                                           
h m 1 1
s  s  1                        s  m  1 f  m i , y i   ds.
m ! 0


 s 
 1 0 
k    1
The integrals                                     ds for various values of k are easily evaluated and are listed in
        k
Table 5.10. For example, when k  3 ,

 s                                   s   s  1  s  2  ds
 1 0   ds   0
3   1                                     1

3                                                        1 2  3

0  s  3s  2s  ds
1 1 3
                2

6
1
1  s4          19 3
   s3  s 2      .
6 4          0 6  4  8
Table 5.10
k                                 0                           1                            2                   3               4                 5

 s                                                     1                         5                      3             251               95
 1 0 
k   1
 ds                     1
       k                                                      2                        12                      8             720               288
As a consequence,
                                                                                     
f  t , y  t   dt  h  f  ti , y  ti    f  ti , y  ti     2 f  ti , y  ti   
ti1                                                   1                          5
 ti
                      2                        12                                    
 (5.28)
hm1 1
0 s  s  1  s  m  1 f i , y i   ds.
 m

m!

Since s  s  1            s  m 1 does not change sign on 0,1 , the Weighted Mean Value Theorem
for Integrals can be used to deduce that for some number  i , where ti 1 m  i  ti 1 , the error

term in Eq. (5.28) becomes

h m1 1
s  s  1              s  m  1 f  m i , y i   ds
m ! 0
h m1 f  m  i , y  i  
 s  s  1  s  m  1 ds
1

m!                   0

or

 s 
hm1 f  m  i , y  i    1
1
0  m  ds.
m
(5.29)
 

Since y  ti 1   y  ti                    f  t , y  t   dt , Eq. (5.26) can be written as
ti1
ti

                                                                                    
y  ti 1   y  ti   h  f  ti , y  ti    f  ti , y  ti     2 f  ti , y  ti   
1                     5
                       2                    12                                      

(5.30)
m 1  s 
 h m 1 f  m   i , y  i    1    ds.
0
 m
EXAMPLE 2
To derive the three-step Adams-Bashforth technique, consider Eq. (5.30) with m  3 :
                                                                           
y  ti 1   y  ti   h  f  ti , y  ti    f  ti , y  ti     2 f  ti , y  ti   
1                        5
                      2                       12                           

 y  ti   h  f  ti , y  ti     f  ti , y  ti    f  ti 1 , y  ti 1   
1
                      2                                                 

 f  ti , y  ti    2 f  ti 1 , y  ti 1    f  ti  2 , y  ti  2    
5
                                                                                        
12                                                                                       
 y  ti    23 f  ti , y  ti    16 f  ti 1 , y  ti 1    f  ti  2 , y  ti  2    .
h
12                                                                                   
The three-step Adams-Bashforth method is, consequently,

w0   , w1  1 , w2   2 ,
h
wi 1  wi            23 f  ti , wi   16 f  ti 1 , wi 1   5 f  ti  2 , wi  2   ,
12                                                                      

for i  2,3,           , N 1 .
Mulitstep methods can also be derived by using Taylor series. An example of the procedure
involved is considered in Exercise 10. A derivation using a Lagrange interpolating polynomial is
discussed in Exercise 9.
The local truncation error for multistep methods is defined analogously to that of one-step
methods. As in the case of one-step methods, the local truncation error provides a measure of how
the different equation fails to solve the difference equation.

Definition 5.15

If y  t  is the solution to the initial-value problem

y '  f t, y  , a  t  b, y  a    ,
and
wi 1  am 1wi  am 2 wi 1                 a0 wi 1m
 h bm f  ti 1 , wi 1   bm1 f  ti , wi  
                                                        b0 f  ti 1m , wi 1m  


is the   i  1 st step in a multistep methods, the local truncation error at this step is
y  ti 1   am 1 y  ti    a0 y  ti 1 m 
 i 1  h  
h                                                         (5.31)
 bm f  ti 1 , y  ti 1    b0 f  ti 1 m , y  ti 1 m    ,
                                                                  

for each i  m  1, m,             , N 1 .                                                                             ■

EXAMPLE 3
To determine the local truncation error for the three-step Adams-Bashforth method derived in
Example 2, consider the form of the error given in Eq. (5.29) and the appropriate entry in Table
5.10:

 s       3h4 3
h4 f 3  i , y  i    1                          f  i , y  i   .
1
0  3   ds 
3

           8

Using the fact that f
3
  , y     y     and the difference equation derived in Example
i       i
4
i

2, we have
y  ti 1   y  ti  1
 i 1  h                        23 f  ti , y  ti    16 f  ti 1 , y  ti 1    5 f  ti  2 , y  ti  2   
h           12                                                                                      
1  3h 4 3                     3h3  4
           f  i , y  i          y  i  , for some i   ti  2 , ti 1  .
h 8                               8
Some of the explicit multistep methods together with their required starting values and local
truncation errors are as follows. The derivation of these techniques is similar to the procedure in
Example 2 and 3.

w0   , w1  1 ,
h                                                                     (5.32)
wi 1  wi        3 f  ti , wi   f  ti 1 , wi 1   ,
2                                       

5
where i  1, 2,            , N  1 . The local truncation error is  i 1  h                         y "'  i  h 2 , for some
12
i  ti1, ti1  .

w0   , w1  1 , w2   2 ,
h                                                                                    (5.33)
wi 1  wi        23 f  ti , wi   16 f  ti 1 , wi 1   5 f  ti  2 , wi  2   ,
12                                                                      

3  4
where i  2,3,             , N  1 . The local truncation error is  i 1  h                        y  i  h3 , for some
8
i   ti 2 , ti 1  .

w0   , w1  1 , w2   2 , w3   3 ,
h                                                                                                    (5.34)
wi 1  wi          55 f  ti , wi   59 f  ti 1 , wi 1   37 f  ti  2 , wi  2   9 f  ti 3 , wi 3   ,
24                                                                                              

251 5
where i  3, 4,            , N  1 . The local truncation error is  i 1  h                      y  i  h4 , for some
720
i   ti 3 , ti 1  .
w0   , w1  1 , w2   2 , w3   3 , w4   4 ,
h
wi 1  wi       1901 f  ti , wi   2774 f  ti 1 , wi 1 
720 
(5.35)

2616 f  ti  2 , wi  2   1274 f  ti 3 , wi 3   251 f  ti  4 , wi  4   ,

95  6
where i  4,5,             N  1 . The local truncation error is  i 1  h                        y  i  h5 , for some
288
i   ti 4 , ti 1  .

Implicit methods are derived by using                  t    i 1                           
, f  ti 1 , y  ti 1   as an additional interpolation node

in the approximation of the integral

f  t , y  t   dt.
ti1
 ti

Some of the more common implicit methods are as follows.

w0   , w1  1 ,
h                                                                              (5.36)
wi 1  wi          5 f  ti 1 , wi 1   8 f  ti , wi   f  ti 1 , wi 1   ,
12                                                               

1  4
where i  1, 2,            N  1 . The local truncation error is  i 1  h                          y  i  h3 , for some
24
i  ti1, ti1  .

w0   , w1  1 , w2   2 ,
h                                                                                                (5.37)
wi 1  wi           9 f  ti 1 , wi 1   19 f  ti , wi   5 f  ti 1 , wi 1   f  ti  2 , wi  2   ,
24                                                                                          

19 5
where i  2,3,             , N  1 . The local truncation error is  i 1  h                        y  i  h4 , for some
720
i   ti 2 , ti 1  .
w0   , w1  1 , w2   2 , w3   3 ,
h
wi 1  wi             251 f  ti 1 , wi 1   646 f  ti , wi 
720 
(5.38)

264 f  ti 1 , wi 1   106 f  ti  2 , wi  2   19 f  ti 3 , wi 3   ,

3  6
where i  3, 4,             N  1 . The local truncation error is  i 1  h                         y  i  h5 , for some
160
i   ti 3 , ti 1  .
It is interesting to compare an m-step Adams-Bashforth explicit method to an (m-1)-step

Adams-Moulton Four-step implicit method. Both involve m evaluations of f per step, and both

have the terms y
 m1
 i  hm   in their local truncation errors. In general, the coefficients of the

terms involving f in the local truncation error are smaller for the implicit methods than for the

explicit methods. This leads to greater stability and smaller roundoff errors for the implicit
methods.

EXAMPLE 4
Consider the initial-value problem

y '  y  t 2  1, 0  t  2, y  0  0.5,
and the approximations given by the explicit Adams-Bashforth four-step method and the implicit
Adams-Moulton three-step method, both using h  0.2 .
The Adams- Bashforth method has the difference equation
h
wi 1  wi           55 f  ti , wi   59 f  ti 1 , wi 1   37 f  ti 2 , wi 2   9 f  ti 3 , wi 3  .
24                                                                                           

for i  3, 4,       ,9 . When simplified using f t, y   y  t 2  1, h  0.2, and ti  0.2i , it

becomes
1
wi 1       35wi  11.8wi 1  7.4wi 2  1.8wi 3  0.192i 2  0.192i  4.736 .
24                                                                   
The Adams-Moulton method has the difference equation
h
wi 1  wi            9 f  ti 1 , wi 1   19 f  ti , wi   5 f  ti 1 , wi 1   f  ti 2 , wi 2   ,
24                                                                                        

for i  2,3,        9 . This reduces to

1
wi 1       1.8wi 1  27.8wi  wi 1  0.2wi 2  0.192i 2  0.192i  4.736  .
24                                                                  

for i  2,3,        9.

The result in Table 5.11 were obtained using the exact values from y  t    t  1  0.5e for
2        t

 , 1 ,  2 , and  3 in the explicit Adams- Bashforth case and for  , 1 , and  2 in the implicit
In Example 4 the implicit Adams- Moulton method gave better results than the explicit Adams-
Bashforth method of the same order. Although this is gererally the case, the implicit methods have
the inherent weakness of first having to convert the method algebraically to an explicit

representation for wi 1 . This procedure is not always possible, as can be seen by considering the

elementary initial-value problem

y '  e y , 0  t  0.25, y  0  1.
Table 5.11

Since f  t , y   e , the three-step Adams-Moulton method has
y

h
wi 1  wi        9e wi1  19e wi  5e wi1  e wi2 
24                                     

as its difference equation, and this equation cannot be solved explicitly for wi 1 .

We could use Newton’s method or the secant method to approximate wi 1 , but this complicates

the procedure considerably. In practice, implicit multistep methods are not used as described
above. Rather, they are used to improve approximations obtained by explicit methods. The
combination of an explicit and implicit technique is called a predictor-corrector method. The
explicit method predicts an approximation, and the implicit method corrects this prediction.
Consider the following fourth-order method for solving an initial-value problem. The first step is

to calculate the starting values w0 , w1 , w2 , and w3 for the four-step explicit Adams-Bashforth
method. To do this, we use a fourth-order one-step method, the Runge-Kutta method of order four.

The next step is to calculate an approximation, w4
 0
, to   y  t4  using the explicit
h
w40  w3 

55 f  t3 , w3   59 f  t2 , w2   37 f  t1 , w1   9 f  t0 , w0   .
24                                                                          
 0
This approximation is improved by inserting w4 in the right side of the three-step implicit

Adams-Moulton method and using that method as a corrector. This gives

w41  w3 
                  h 
24                  
9 f t4 , w40  19 f  t3 , w3   5 f  t2 , w2   f  t1 , w1  .


The only new function evaluation required in this procedure is f t4 , w4                                0
   in the corrector

equation; all the other values of f have been calculated for earlier approximations.

The value w4 is then used as the approximation to y  t4  , and the technique of using the
1

Adams-Bashforth method as a predictor and the Adams-Moulton method as a corrector is repeated

to find w5 and w5 , the initial and final approximations to y  t5  , etc.
 0            1

Improved approximations to y ti 1  could be obtained by iterating the Adams-Moulton formula

wi11  wi 
k                 h 
24 

9 f ti 1 , wi1  5 f  ti 1 , wi 1   f  ti 2 , wi 2  .
k
                                            
However,     w   converges to the approximation given by the implicit formula rather than to
k 1
i 1

the solution y ti 1  , and it is usually more efficient to use a reduction in the step size if

improved accuracy is needed.
Algorithm 5.4 is based on the fourth-order Adams-Bashforth method as predictor and one iteration
of the Adams-Moulton method as corrector, with the starting values obtained from the
fourth-order Runge-Kutta method.

Purpose: To approximate the solution of the initial-value problem

y '  f t, y  , a  t  b, y  a    ,

at    N  1 equally spaced numbers in the interval  a, b :
INPUT       endpoints a, b ; integer N ; initial condition  .

OUTPUT approximation w to y at the                             N  1      values of t .

Step 1      h  b  a  N ;
t0  a ;

w0   ;

OUTPUT        t0 , w0  .
Step 2 For i  0,1, 2 , do Step 3-4

(*Computing starting values using Runge-Kutta method*.)

Step 3       K1  hf ti , wi  ;

K2  hf ti  h 2, wi  K1 2 ;

K3  hf  ti  h 2, wi  K2 2 ;

K4  hf ti  h, wi  K3  .

wi 1  wi   K1  2K2  2K3  K4  6 ;

ti 1  a  (i  1)h .

Step 4

Step 5 For i  3,      , N  1 do Steps 6-8

Step 6        t 0  a  ( i  1 )h
;

w0  wi  h 55 f  ti , wi   59 f  ti 1 , wi 1   37 f  ti 2 , wi 2 

9 f  ti 3 , wi 3  24;
             Predict      wi 1 

w0  wi  h 9 f  t 0, w0   19 f  ti , wi   5 f  ti 1 , wi 1 

 f  ti 2 , wi 2   24.
            Correct       wi 1 
Step 7
C             PREPARE FOR NEXT ITERATION
T(I+1) = TO
W(I+1) = WO .
Step 8
Step 9 RETURN.

This produced ADAMS-FOURTH ORDER PREDICTOR-CORRECTOR method described in the
following subroutine:
C**********************************************************************
C ADAMS-FOURTH ORDER PREDICTOR-CORRECTOR ALGORITHM 5.4                  *
C                                                                      *
C***********************************************************************
C     TO APPROXIMATE THE SOLUTION OF THE INITIAL VALUE PROLEM:
C               Y'=F(T,Y), A<=T<=B, Y(A)=ALPHA,
C     AT (N+1) EQUALLY SPACED NUMBERS IN THE INTERVAL [A,B].
C
C     INPUT: ENDPOINTS A,B; INITIAL CONDITION ALPHA; INTEGER N.
C
C     OUTPUT: APPROXIMATION W TO Y AT THE (N+1) VALUES OF T.
C**********************************************************************

INTEGER N
REAL A,B,ALPHA
REAL T(0:10),W(0:10)
EXTERNAL F

C  STEP 1
H = (B-A)/N
T(0) = A
W(0) = ALPHA
C STEP 2
DO 110 I=0,2
C     STEPS 3-4
C     COMPUTE STARTING VALUES USING RUNGE-KUTTA METHOD GIVEN IN A
C     SUBROUTINE--NOTE: FUNCTION F IS NEEDED IN THE SUBROUTINE
CALL NRK4(H,T(I),W(I),T(I+1),W(I+1))
C         CALL RK4(1,T(I),T(I)+H,W(I),T(I+1),W(I+1))
C         STEP 4,
110 CONTINUE
C STEP 5
DO 20 I=3,N-1
C          STEP 6-8
C          STPE 6
C          TO, WO WILL BE USED IN PLACE OF T, W RESP.
TO=A+(I+1)*H
C          PREDICT W(I+1)
WO = W(I)+H*(55*F(T(I),W(I))-59*F(T(I-1),W(I-1))
*        +37*F(T(I-2),W(I-2))-9*F(T(I-3),W(I-3)))/24
C          CORRECT W(I)
WO = W(I)+H*(9*F(TO,WO)+19*F(T(I),W(I))
*        -5*F(T(I-1),W(I-1))+F(T(I-2),W(I-2)))/24
C          STEP 7
C          PREPARE FOR NEXT ITERATION
T(I+1) = TO
W(I+1) = WO
STEP 8
20      CONTINUE
C       STEP 9
RETURN
END

SUBROUTINE NRK4(H,TO,WO,TI,WI)
TI = TO+H
XK1 = H*F(TO,WO)
XK2 = H*F(TO+H/2,WO+XK1/2)
XK3 = H*F(TO+H/2,WO+XK2/2)
XK4 = H*F(TI,WO+XK3)
WI = WO+(XK1+2*(XK2+XK3)+XK4)/6
RETURN
END
■
EXAMPLE 5 Use Adams Fourth-order Predictor-Corrector method to approximate the solution
of initial-value problem

y '  y  t 2  1, 0  t  2, y  0  0.5,
with N  10 . AL54.f is shown as follows:
C******************************************************************
C Example 5 (Section 5.6) Using ADAMS-FOURTH ORDER
C              PREDICTOR-CORRECTOR ALGORITHM (ORDER 4) to
C             approximete the solution of the initial-value problem
C              dy/dt=y-t^2+1, y(0)=0.5
C******************************************************************
C
INTEGER N
REAL A,B,ALPHA
REAL T(0:10),W(0:10),Y(0:10)
INTRINSIC EXP,ABS

OPEN(UNIT=10,FILE='AL54.doc',STATUS='UNKNOWN')

C    *** Input initial values

A = 0.0
B = 2.0
ALPHA = 0.5
N = 10

WRITE(10,*) ' T(i)         W(i)          Y(i)      |W(i)-Y(i)|'
WRITE(10,*) '---------------------------------------------'
WRITE(*,*) ' T(i)         W(i)             Y(i)        |W(i)-Y(i)|'
WRITE(*,*) '---------------------------------------------'

DO 10 I = 0,10
C *** Exact solution
Y(I) = (T(I)+1.0)*(T(I)+1.0)-0.5*EXP(T(I))
WRITE(10,99) T(I),W(I),Y(I),ABS(W(I)-Y(I))
WRITE(*,99) T(I),W(I),Y(I),ABS(W(I)-Y(I))
10     CONTINUE
STOP
99     FORMAT(1X,F5.1,3F12.7)
END

REAL FUNCTION F(T,Y)
C=========================================
C PURPOSE
C Find the value of function f(t,y) = y-t^2+1
C--------------------------------------------------------------------
C
REAL T,Y
F = Y-T*T+1.0
RETURN
END

Table 5.12 lists the results obtained by using Algorithm 5.4 for the initial-value problem

y '  y  t 2  1, 0  t  2, y  0  0.5,
with N  10 . The results here are more accurate than those in Example 4, which used only the
corrector (that is, the implicit Adams-Moulton method), but this is not always the case.

Table 5.12
Error

ti        yi  y ti                wi                 yi  wi
0.0        0.5000000             0.5000000                 0
0.2        0.8292986             0.8292933             0.0000053
0.4        1.2140877             1.2140762             0.0000114
0.6        1.6489406             1.6489220             0.0000186
0.8        2.1272295             2.1272056             0.0000239
1.0        2.6408591             2.6408286             0.0000305
1.2        3.1799415             3.1799026             0.0000389
1.4        3.7324000             3.7323505             0.0000495
1.6         4.2834838               4.2834208                0.0000630
1.8         4.8151763               4.8150964                0.0000799
2.0         5.3054720               5.3053707                0.0001013

Other multistep methods can be derived using integration of interpolating polynomials over

intervals of the form t j , t j 1  , for j  i  1 , to obtain an approximation to y ti 1  . When an
             

interpolating polynomial is integrated over ti 3 , ti 1  , the result is the explicit Milne’s method:

4h
 2 f  ti , wi 1   f  ti 1 , wi 1   2 f  ti 2 , wi 2   .
wi 1  wi 3 
3                                                                   
14 4 5
which has local truncation error     h y i  , for some i  ti 3 , ti 1  .
45
This method is occasionally used as a predictor for the implicit Simpson’s method.
h
wi 1  wi 1   f  ti 1 , wi 1   4 f  ti , wi   f  ti 1 , wi 1   ,
3                                                             

which has local truncation error  h4 90 y                 5
i  , for some i  ti1, ti1  , and is obtained
by integrating an interpolating polynomial over ti 1 , ti 1  .

The local truncation error involved with a predictor-corrector method of the Milne-Simpson type
is generally smaller than that of the Adams-Bashforth-Moulton method. But the technique has
limited use because of roundoff error problems, which do not occur with the Adams procedure.
Elaboration on this difficulty is given in Section 5.10.

EXERCISE SET 5.6
Q4. Use Adams Fourth-order Predictor-Corrector method to approximate the solution of
initial-value problem

(a)     y '  te3t  2 y, 0  t  1, y  0  0.0,                   with     N  10 ;          The        exact   solution

1      1     1
y  t   te3t  e3t  e2t and compare the results to the actual values.
5      25    25

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