A MATHEMATICA interface for the Taylor series method

A MATHEMATICA interface for theTaylor series method

A. Abad, R. Barrio, F. Blesa, M. Rodrıguez

Grupo de Mecanica Espacial. University of Zaragoza

A M ATHEMATICA interface for the Taylor series method – p. 1/21

Taylor Series Method (I):• Differential equation:

x = f(t,x,p), x(0) = x0



x = f(t,x,p), x(0) = x0

• Power series expansion solution

x(t) =∑

i

xiti



x = f(t,x,p), x(0) = x0

• Power series expansion solution

x(t) =∑

i

xiti

• Formn-th to (n + 1)-th order

f(t,∑

i

xiti,p) =

∑

i

fiti =

∑

i

(i + 1)xi+1ti, xi+1 =

fi

i + 1


Taylor Series Method (II):• OrderN of the method:

• − log ǫ/2, ǫ =tolerance• Variable order strategy (Barrio et al.).




• Step (convergence radius * safety factor)

h = min

(

ǫ1/(n+1)

‖xn ‖1/n∞

,ǫ1/n

‖xn−1 ‖1/(n−1)∞

)




• Step (convergence radius * safety factor)

h = min

(

ǫ1/(n+1)

‖xn ‖1/n∞

,ǫ1/n

‖xn−1 ‖1/(n−1)∞

)

• Computing:x0 = x(t0), . . . xi = x(ti),xi+1 = x(ti+1), . . .

• ti+1 < ti + h: direct evaluation of power series

• ti+1 > ti + h: recomputation of power series takingti + h

as initial point.


Sensitivity analysis• Differential equation: x = f(t,x,p), x(0) = x0



• Partial derivativessk = ∂x/∂pk, with respect to a parameter (or

an initial condition) . Sensitivity equations:

sk =∂f

∂xsk +

∂f

∂pk





sk =∂f

∂xsk +

∂f

∂pk

• (skj)m = ∂2xm/∂pk∂pj . Higher order sensitivity equations:

d(skj)mdt

=∂fm

∂xskj+(sj)

T ∂2fm

∂x2sk+

∂2fm

∂pj∂xsk+

∂2fm

∂pk∂xsj+

∂2fm

∂pk∂pj





sk =∂f

∂xsk +

∂f

∂pk

• (skj)m = ∂2xm/∂pk∂pj . Higher order sensitivity equations:

d(skj)mdt

=∂fm

∂xskj+(sj)

T ∂2fm

∂x2sk+

∂2fm

∂pj∂xsk+

∂2fm

∂pk∂xsj+

∂2fm

∂pk∂pj

• Extended Taylor method:easier way to compute partial

derivatives up to any orderp.


Structure of functions

f(x) =sin x2

x2 + 5



f(x) =sin x2

x2 + 5

Tree structure :

/

sin

*

x x

+

5 *

x x



f(x) =sin x2

x2 + 5

Tree structure :

/

sin

*

x x

+

5 *

x x

Chain structure(linked functions):

l1 = x

l2 = l1 l1l3 = 5 + l2l4 = sin l2l5 = l4/l3f(x) = l5


Linked functions, useful for:• Efficient evaluation.



• Automatic differentiation.

l1 = x

l2 = l1 l1

l3 = 5 + l2

l4 = sin l2

l5 = l4/l3

f(x) = l5

l′1 = 1

l′2 = 2 l1 l′1

l′3 = l′2

l′4 = l′2 cos l2

l′5 = (l′4l3 − l4l′3)/(l3 l3)

f ′(x) = l′5




l1 = x

l2 = l1 l1

l3 = 5 + l2

l4 = sin l2

l5 = l4/l3

f(x) = l5

l′1 = 1

l′2 = 2 l1 l′1

l′3 = l′2

l′4 = l′2 cos l2

l′5 = (l′4l3 − l4l′3)/(l3 l3)

f ′(x) = l′5

• Power series: Algebra of Series & Taylor

method.




l1 = x

l2 = l1 l1

l3 = 5 + l2

l4 = sin l2

l5 = l4/l3

f(x) = l5

l′1 = 1

l′2 = 2 l1 l′1

l′3 = l′2

l′4 = l′2 cos l2

l′5 = (l′4l3 − l4l′3)/(l3 l3)

f ′(x) = l′5

• Power series: Algebra of Series & Taylor method.• Partial derivatives: Extended Taylor method.


Algebra of series (I)

u =n∑

i=0

uiti, v =

n∑

i=0

viti, w =

n∑

i=0

witi, α ∈ IR

Elementary functions

w = u + v → wk = uk + wk,

w = uv → wk =∑k

j=0 ujvk−j,

w = α + u → w0 = α + u0, wk = uk,

w = αu → wk = αuk,

w = 1/u → w0 = − 1u0

, wk = − 1u0

∑k−1j=0 ujvk−j,

w = dudt → wk = (k + 1)uk.


Algebra of series (II)

w = f(u), withdf

du=

1

g(u)−→ wk =

1

go

uk − 1

k

k−1∑

j=1

jwjgk−j

• f = log u,−→ g = u.

• f = asin u,−→ g =√

1 − u2.

• f = acos u,−→ g = −√

1 − u2.

• f = atan u,−→ g = 1 + u2.

• f = asinhu,−→ g =√

1 + u2.

• f = acoshu,−→ g =√

u2 − 1.

• f = atanhu,−→ g = 1 − u2.


Algebra of series (III)More functions:

• uα, αu, uv

• s = sin u, c = cosu

sk =1

k

k−1∑

j=0

(k − j)cjuk−j , ck = −1

k

k−1∑

j=0

(k − j)sjuk−j .

• s = sinh u, c = cosh u

sk =1

k

k−1∑

j=0

(k − j)cjuk−j , ck =1

k

k−1∑

j=0

(k − j)sjuk−j .

• tanu =sin u

cos u, tanh u =

sinh u

coshu


Taylor Series Method (TSM)l1 = x

l2 = l1 f1

l3 = 5 + l2

l4 = sin l2

l5 = l4/l3



l2 = l1 f1

l3 = 5 + l2

l4 = sin l2

l5 = l4/l3

l1 = x

l2 = l1 l1

l3 = 5 + l2

l4 = sin l2

l5 = cos l2

l6 = l4/l3



l2 = l1 f1

l3 = 5 + l2

l4 = sin l2

l5 = l4/l3

l1 = x =

n∑

i=0

xiti

l2 = l1 l1

l3 = 5 + l2

l4 = sin l2

l5 = cos l2

l6 = l4/l3 =

n∑

i=0

fiti

xn+1 = fn/(n + 1)

Algebra series + extended linked functions:Taylor series method

x0, . . . , xn → f(t,

n∑

i=0

xiti, p) =

n∑

i=0

fiti → xn+1 =

fn

n + 1


Partial derivatives: Notation(I)• f(x),x = (x1, . . . , xn) ∈ Rn

fi=

∂(i1+...+in)f

∂xi11 . . . ∂xin

n

, i = (i1, . . . , in) ∈ Nn0



fi=

∂(i1+...+in)f

∂xi11 . . . ∂xin

n

, i = (i1, . . . , in) ∈ Nn0

• then-tuplei identifies each partial derivative.



fi=

∂(i1+...+in)f

∂xi11 . . . ∂xin

n

, i = (i1, . . . , in) ∈ Nn0

• then-tuplei identifies each partial derivative.• i − j = (i1 − j1, . . . , in − jn).



fi=

∂(i1+...+in)f

∂xi11 . . . ∂xin

n

, i = (i1, . . . , in) ∈ Nn0

• then-tuplei identifies each partial derivative.• i − j = (i1 − j1, . . . , in − jn).• i∗ = (0, . . . , 0, ik − 1, . . . , in) whereik is the first

nonzero element.



fi=

∂(i1+...+in)f

∂xi11 . . . ∂xin

n

, i = (i1, . . . , in) ∈ Nn0

• then-tuplei identifies each partial derivative.• i − j = (i1 − j1, . . . , in − jn).• i∗ = (0, . . . , 0, ik − 1, . . . , in) whereik is the first

nonzero element.• Dm

n = {i | i1 + . . . + in ≤ m} is the set of allpartial derivatives until orderm


Partial derivatives: Notation(II)• Total order inDm

n : i ≤ j



n : i ≤ j

• V(i) = {v |v ≤ i}, is the set of all partialderivativesv that we need to compute thederivativei.



n : i ≤ j


• V(D) =⋃

i∈D V(i) ⊂ Dmn , is the set of all partial

derivativesv that we need to compute thederivative of the setD.



n : i ≤ j


• V(D) =⋃

i∈D V(i) ⊂ Dmn , is the set of all partial

derivativesv that we need to compute thederivative of the setD.

• O(V(D)) is the result of ordering the setV(D)


Partial derivatives:Notation(III)

f [j](t) =1

j!f (j)(t), f [j]

i=

∂(i1+...+in)f [j]

∂xi11 . . . ∂xin

n

,

(

i

v

)

=

(

i1v1

)

. . .

(

invn

)


Partial derivatives:Notation(III)

f [j](t) =1

j!f (j)(t), f [j]

i=

∂(i1+...+in)f [j]

∂xi11 . . . ∂xin

n

,

(

i

v

)

=

(

i1v1

)

. . .

(

invn

)

• h(t) = exp(f(t))

h[0]0

= exp(f [0](t))

h[0]i

=∑

v∈V(i∗)

(

i∗

v

)

h[0]v

f [0]i−v

h[n]i

= 1n

n−1∑

j=0

(n − j)

∑

v∈V(i)

(

i

v

)

h[j]v

f [n−j]i−v


Generating partial derivatives

∑

v,w∈V(i)

Ci,vf [−]v

g[−]w

,∑

v,w∈V(i∗)

Ci∗,vf [−]v

g[−]w

, w = i−v, Ci,v =

(

i

v

)

,

• Compute theN elements ofV(D).



∑

v,w∈V(i)

Ci,vf [−]v

g[−]w

,∑

v,w∈V(i∗)

Ci∗,vf [−]v

g[−]w

, w = i−v, Ci,v =

(

i

v

)

,


• SortV(D) −→ O(V(D)) −→ 0 ≤ i < N .



∑

v,w∈V(i)

Ci,vf [−]v

g[−]w

,∑

v,w∈V(i∗)

Ci∗,vf [−]v

g[−]w

, w = i−v, Ci,v =

(

i

v

)

,


• SortV(D) −→ O(V(D)) −→ 0 ≤ i < N .

• i ∈ O(V(D)) −→ V(i),V(i∗) −→ Ci,v, Ci∗,v



∑

v,w∈V(i)

Ci,vf [−]v

g[−]w

,∑

v,w∈V(i∗)

Ci∗,vf [−]v

g[−]w

, w = i−v, Ci,v =

(

i

v

)

,


• SortV(D) −→ O(V(D)) −→ 0 ≤ i < N .

• i ∈ O(V(D)) −→ V(i),V(i∗) −→ Ci,v, Ci∗,v

• Two lists of integer numbers that represent the indexes

v,w ∈ V(i) .



∑

v,w∈V(i)

Ci,vf [−]v

g[−]w

,∑

v,w∈V(i∗)

Ci∗,vf [−]v

g[−]w

, w = i−v, Ci,v =

(

i

v

)

,


• SortV(D) −→ O(V(D)) −→ 0 ≤ i < N .

• i ∈ O(V(D)) −→ V(i),V(i∗) −→ Ci,v, Ci∗,v


v,w ∈ V(i) .


v,w ∈ V(i∗) .


Storage data structures• Example: ∂3f

∂x1∂x2

2



∂x1∂x2

2

• O(V(D)) = {(0, 0), (1, 0), (0, 1), (1, 1), (0, 2), (1, 2)}= {0, 1, 2, 3, 4, 5}



∂x1∂x2

2

• O(V(D)) = {(0, 0), (1, 0), (0, 1), (1, 1), (0, 2), (1, 2)}= {0, 1, 2, 3, 4, 5}

• V(4) = V((0, 2)) = {(0, 0), (0, 1), (0, 2)} = {0, 2, 4}



∂x1∂x2

2

• O(V(D)) = {(0, 0), (1, 0), (0, 1), (1, 1), (0, 2), (1, 2)}= {0, 1, 2, 3, 4, 5}

• V(4) = V((0, 2)) = {(0, 0), (0, 1), (0, 2)} = {0, 2, 4}• Data arrays:

PREV_ACUM 0 1 3 5 9 12 18

PREV_COEF 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 1 2 1

PREV_VI 0 0 1 0 2 0 2 1 3 0 2 4 0 2 4 1 3 5

PREV_IV 0 1 0 2 0 3 1 2 0 4 2 0 5 3 1 4 2 0


TIDES (I)• TIDES :a Taylor series Integrator for Differential EquationS.



• Numerical software extremely easy to use via a MATHEMATICA

preprocessor.




preprocessor.

• Taylor series method using Variable-Stepsize Variable-Order

formulation and extended formulas for the variational equations.




preprocessor.



• Automatic construction of Fortran codes for solving solutions of

ODEs.




preprocessor.




ODEs.

• Automatic construction of C codes for solving solutions of ODEs

and variational equationsup to any order (and sensitivities with

respect to any parameter up to any order) .




preprocessor.




ODEs.

• Automatic construction of C codes for solving solutions of ODEs

and variational equationsup to any order (and sensitivities with

respect to any parameter up to any order) .

• Easy to usearbitrary precision solver(do you need 500 digits?,

1000?)


TIDES (II)MathTIDES preprocessor MATHEMATICA

libTIDES.a library C

(objects or source code)

minf-tides basic TSM FORTRAN files

minc-tides basic TSM C files

dp-tides extended TSM C files

basic + partial derivatives + libTIDES.a

mp-tides extended TSM C files

extended + arbitrary precision+ libTIDES.a

+ libmpfr.a


Double precision tests


Multiple precision test


TIDES session


Join TIDES community• Free software• Where?:http://gme.unizar.es/software/tides• or email: [email protected]


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A MATHEMATICA interface for the Taylor series method