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On a modification of Newton's methods for the numerical solution of boundary problems
ON A MODIFICATIONOF NEWTON'S METEODS FOR TEE NUMERICAL SOLUTION OF BOUNDARY PROBLEMS* V. K.
ISAYEV
and
V.V.
SONIN
(MOSCOW) (Received
1 April
1963)
A large number of problems of optimal control using L.S. Pontryagin’ 6 principle of the maximum lead to the solution of boundary value problems. In this case if the system of equations of optimal motion can be integrated the boundary value problem reduces to a system of algebraic or transcendental equations. OtherwIse the solution of the boundary value problem Itself reduces to the successive solution of Caachy problems. In both case8 in the process of solving the boundary value problem we have to carry out operations, using finite formulae or numerical lntegration. which provide In the first case the finite posltion of a point in extended phase space and In the second case the whole trajectory. (In the extended phase space we use as coordinates phase coordinates, the conjugate variables and the time t. Without loss of generality we shall assume below that the segment 10, ~1 is fixed. If not there is a supplementary condition to determine it and the integration limits are found from any of the boundary conditions.) It is clear from this that the basic problem consists in determining (or concluding the determination of) the initial pofnt In the extended phase space such that the trajectory which starts from It time t = T satisfies all the boundary conditions. To solve the boundary problems we use Newton’s method and this often diverges If the first approximation is not sufflclentlg good. Let us turn to the simplest exsstple. It is required to find the zero of a COBtinaous piece-wise smooth function of one variable (Pig. 1):
l
Zh.
Vych. Afat.
3, No.
6,
1114 - 1116. 1525
1963.
V.K.
1528
Isaycv
and
V. V.
Soning
f -++3. Y (4 = t -2z+13,
0)
Y 3
2 I 0 -I
0
-2 -3 FIO.
This alearly
example dlrplys
As our first
FIO. 2.
1.
belongs to the first the features which approxlmatlon
type of problem interest us.
we take
XQ = 5. Using
=&+I= Zk + i3k= Z&we find
the
sequence q = 9,
The cyclical shows that It It
of subsequent 1FI= 2,
the
obvious
above and
formula
Y @J dY (~~11 h
(2)
approrlmationa: zr-
9,
z.4=2,....
nature of the iterative process is at once Is u8elese to continue with rule (2).
la easy to find oases where the doomed to failure: the iterative neighbourhood of the required solution alao
described
evident
and
attempt to use Newton*tv method proaess either cycles In the or diverges.
is
The reason for failure with example (1) is that the magnitude of the step e& in the separate stages is too large, and is due to the fact that Newton’s method only uses information about the Initial point of the iteration. which essentially gives a representation By using this information, the function in a small nelghbourhood of the initial point, the posltion of the zero Is forecast and the next step Is made in the direction found. of
Yodi jication
of
However the accuracy based decreases
o j Ncvtorn’r
expmsioa
Taylor
the
as the distance
1527
aethodr
on vhich
from the Initial
point
formula
(2) is
increases.
In certain cases to overcome this ve oao lntroduoe a correction at certain stages of the process. Of course this does not eliminate the need for
additional
information
though the 8mount required
about the operation
must be minimised
of the process
al-
as much 8s possible.
Let us introduce size
a correction depending on the distribution of the w.r.t. the bound8rY conditions at discrete points of
of the error
a rax joining function
two successive
Here k Is the index of which increases length
of
linearly
the k-th
one-dirrensional formula steps
(2)
criterion Wk.
@(k,
for lewton*s 11,
echeae
convergence
In prsctice
1.1)
the error
SStiSfied
(k -
l)-th
1).
If
k-th
scheme,
and the k-th the divergence
step,
i.e.
calculating
from the point a8
the
a!
view of the
yet unsolved. universal. of the ray.
of the discrete
8 98r8bol8
a) and find
Choosing
If
srgument
through the three
the minimum point
3(b (k, 0) - 44,(k, +) +
for
convexity
2@ (k, f)
points
eXr of the
ray,
c; H 8nd so on. When ai hss been instead
of
@ fk, i)
(3)
+ Q, (k, 1) ] *
below is not 88tisfled
we bsve described
this
Sk Is the
points
operations
Applying
X&l for the
cD(k, $41 St the centre
4 [‘I’ 6, 0) -
=kCl
[%&_I,
pollpomial:
a* =
z,+a*sk
xkl;
For a = 0 8nd a = 1
St the
1) form a function
to @(k,
interpolsting
0
(O
[X&-I,
letbod is sufficiently
convex below we cOn8truCt
the condition
a parameter
of
and @(k,
ray which is optimal
the following
which approximates
=
the %1ss*
of tbe ray [S&_I, X&l. !fbe N& and of the positions oj, j = 1, . . . , N& of
Qt(k, 0). Q(k. $1 and O(k,
Lsgrsnge
(2)). for the
of the method is a complex problem,
a = % we cslculate which is
is
at one or several
on this
length
supplement the classical
we
the ooantftx
Of
po’ints
If
Consider
of the ray
(the
errors
01 = @(I -
a is
step,
given by foxaula
the following
of the error
choice
the iteration
of Iteration
case is
gives
01 (Q(k,
these
of the iteration.
with the length
step
respectfrelx:
value
points
(error),
me repested
or ti CO
but now on half
found we use the relstfon
(2).
method to exaaple
(1)
we obtain
(Fig-
2).
the
of the initial
1528
V. K. Isuyev
mad V. V. Swing
aqi+1,
Y(z,+f%)=Y(7)=-1,
and the process
is
e, =--
L
tioa
by a p8r8bOl8
;
CC*=
2~; f
then rapidly e3 E
10 6 --1l
ar*+,
completed with the classical
A =
6 f ; CD
(2, 1) .= 0.
scheme
Obviously
is not the only method of correction
approxima-
which depends on
the results of exploring the *inside of the step*. The following simple algorithm has also proved to be userul in practice: (1)
we carry
@(k, I); (3)
if
out a ‘classical
step*;
(2)
we calculate
O(k. 1) < (o(k, 0) the cl8sslc8l
otherwise
$(k. !4) is calculated;
scheme is
again used;
(4)
otherwise
if
Wk.
!4) is
@(k.
(ofk, 0) and
scheme is 35) < @(k.
round.
very
repeated;
0) Newtoa’s
and so oa.
8ad P
=
1, %. %. . . . . A similar explaiaed
method for
in
This modification culty
is
the roots
of Newton’s
to the multi-dimensional
pore. it
finding
of aa algebraic
equation
In the solution
advisable
to bring
method can be extended without diffi-
case.
of boundary value problems of the second kind the equations
to dimensioalees
fore
so that
values of the extended phase coordinates are of the same order; is important to remember that as the accuracy of the calculation trajectory
is
is
(11.
reduced the process
also
the
it
of the
may diverge.
The authors have been working on methods of improving the convergence of Newtoa’s method for the aumerical solution of two-point boundary value probleas
used has
or high order (10th - 14th) since 1981; the number of psrsmters ta the method has reached six. This aodif~cation of Newton’s method also been tested at the Computing Centre of the USSR Academy of
Sciences
by V.N.
Lebedev. Translated
by R. Peinsteia
REFERRNCE 1.
tans,
J.N. .Nurcrical
acthods
for
High-Speed
metody dlya bistrodeistvuyushchikh 90 in.
lit‘.
* MOBCOW.
1982.
Coaputers
vychislftel’nykh
(Chfsleaaye mashin).
Izd-
ISAYEV
and
V.V.
SONIN
(MOSCOW) (Received
1 April
1963)
A large number of problems of optimal control using L.S. Pontryagin’ 6 principle of the maximum lead to the solution of boundary value problems. In this case if the system of equations of optimal motion can be integrated the boundary value problem reduces to a system of algebraic or transcendental equations. OtherwIse the solution of the boundary value problem Itself reduces to the successive solution of Caachy problems. In both case8 in the process of solving the boundary value problem we have to carry out operations, using finite formulae or numerical lntegration. which provide In the first case the finite posltion of a point in extended phase space and In the second case the whole trajectory. (In the extended phase space we use as coordinates phase coordinates, the conjugate variables and the time t. Without loss of generality we shall assume below that the segment 10, ~1 is fixed. If not there is a supplementary condition to determine it and the integration limits are found from any of the boundary conditions.) It is clear from this that the basic problem consists in determining (or concluding the determination of) the initial pofnt In the extended phase space such that the trajectory which starts from It time t = T satisfies all the boundary conditions. To solve the boundary problems we use Newton’s method and this often diverges If the first approximation is not sufflclentlg good. Let us turn to the simplest exsstple. It is required to find the zero of a COBtinaous piece-wise smooth function of one variable (Pig. 1):
l
Zh.
Vych. Afat.
3, No.
6,
1114 - 1116. 1525
1963.
V.K.
1528
Isaycv
and
V. V.
Soning
f -++3. Y (4 = t -2z+13,
0)
Y 3
2 I 0 -I
0
-2 -3 FIO.
This alearly
example dlrplys
As our first
FIO. 2.
1.
belongs to the first the features which approxlmatlon
type of problem interest us.
we take
XQ = 5. Using
=&+I= Zk + i3k= Z&we find
the
sequence q = 9,
The cyclical shows that It It
of subsequent 1FI= 2,
the
obvious
above and
formula
Y @J dY (~~11 h
(2)
approrlmationa: zr-
9,
z.4=2,....
nature of the iterative process is at once Is u8elese to continue with rule (2).
la easy to find oases where the doomed to failure: the iterative neighbourhood of the required solution alao
described
evident
and
attempt to use Newton*tv method proaess either cycles In the or diverges.
is
The reason for failure with example (1) is that the magnitude of the step e& in the separate stages is too large, and is due to the fact that Newton’s method only uses information about the Initial point of the iteration. which essentially gives a representation By using this information, the function in a small nelghbourhood of the initial point, the posltion of the zero Is forecast and the next step Is made in the direction found. of
Yodi jication
of
However the accuracy based decreases
o j Ncvtorn’r
expmsioa
Taylor
the
as the distance
1527
aethodr
on vhich
from the Initial
point
formula
(2) is
increases.
In certain cases to overcome this ve oao lntroduoe a correction at certain stages of the process. Of course this does not eliminate the need for
additional
information
though the 8mount required
about the operation
must be minimised
of the process
al-
as much 8s possible.
Let us introduce size
a correction depending on the distribution of the w.r.t. the bound8rY conditions at discrete points of
of the error
a rax joining function
two successive
Here k Is the index of which increases length
of
linearly
the k-th
one-dirrensional formula steps
(2)
criterion Wk.
@(k,
for lewton*s 11,
echeae
convergence
In prsctice
1.1)
the error
SStiSfied
(k -
l)-th
1).
If
k-th
scheme,
and the k-th the divergence
step,
i.e.
calculating
from the point a8
the
a!
view of the
yet unsolved. universal. of the ray.
of the discrete
8 98r8bol8
a) and find
Choosing
If
srgument
through the three
the minimum point
3(b (k, 0) - 44,(k, +) +
for
convexity
2@ (k, f)
points
eXr of the
ray,
c; H 8nd so on. When ai hss been instead
of
@ fk, i)
(3)
+ Q, (k, 1) ] *
below is not 88tisfled
we bsve described
this
Sk Is the
points
operations
Applying
X&l for the
cD(k, $41 St the centre
4 [‘I’ 6, 0) -
=kCl
[%&_I,
pollpomial:
a* =
z,+a*sk
xkl;
For a = 0 8nd a = 1
St the
1) form a function
to @(k,
interpolsting
0
(O
[X&-I,
letbod is sufficiently
convex below we cOn8truCt
the condition
a parameter
of
and @(k,
ray which is optimal
the following
which approximates
=
the %1ss*
of tbe ray [S&_I, X&l. !fbe N& and of the positions oj, j = 1, . . . , N& of
Qt(k, 0). Q(k. $1 and O(k,
Lsgrsnge
(2)). for the
of the method is a complex problem,
a = % we cslculate which is
is
at one or several
on this
length
supplement the classical
we
the ooantftx
Of
po’ints
If
Consider
of the ray
(the
errors
01 = @(I -
a is
step,
given by foxaula
the following
of the error
choice
the iteration
of Iteration
case is
gives
01 (Q(k,
these
of the iteration.
with the length
step
respectfrelx:
value
points
(error),
me repested
or ti CO
but now on half
found we use the relstfon
(2).
method to exaaple
(1)
we obtain
(Fig-
2).
the
of the initial
1528
V. K. Isuyev
mad V. V. Swing
aqi+1,
Y(z,+f%)=Y(7)=-1,
and the process
is
e, =--
L
tioa
by a p8r8bOl8
;
CC*=
2~; f
then rapidly e3 E
10 6 --1l
ar*+,
completed with the classical
A =
6 f ; CD
(2, 1) .= 0.
scheme
Obviously
is not the only method of correction
approxima-
which depends on
the results of exploring the *inside of the step*. The following simple algorithm has also proved to be userul in practice: (1)
we carry
@(k, I); (3)
if
out a ‘classical
step*;
(2)
we calculate
O(k. 1) < (o(k, 0) the cl8sslc8l
otherwise
$(k. !4) is calculated;
scheme is
again used;
(4)
otherwise
if
Wk.
!4) is
@(k.
(ofk, 0) and
scheme is 35) < @(k.
round.
very
repeated;
0) Newtoa’s
and so oa.
8ad P
=
1, %. %. . . . . A similar explaiaed
method for
in
This modification culty
is
the roots
of Newton’s
to the multi-dimensional
pore. it
finding
of aa algebraic
equation
In the solution
advisable
to bring
method can be extended without diffi-
case.
of boundary value problems of the second kind the equations
to dimensioalees
fore
so that
values of the extended phase coordinates are of the same order; is important to remember that as the accuracy of the calculation trajectory
is
is
(11.
reduced the process
also
the
it
of the
may diverge.
The authors have been working on methods of improving the convergence of Newtoa’s method for the aumerical solution of two-point boundary value probleas
used has
or high order (10th - 14th) since 1981; the number of psrsmters ta the method has reached six. This aodif~cation of Newton’s method also been tested at the Computing Centre of the USSR Academy of
Sciences
by V.N.
Lebedev. Translated
by R. Peinsteia
REFERRNCE 1.
tans,
J.N. .Nurcrical
acthods
for
High-Speed
metody dlya bistrodeistvuyushchikh 90 in.
lit‘.
* MOBCOW.
1982.
Coaputers
vychislftel’nykh
(Chfsleaaye mashin).
Izd-