DIFFERENTIATION AND INTEGRATION BY USING MATRIX

Journal of Applied Mathematics and Computational Mechanics 2014, 13(2), 63-71
DIFFERENTIATION AND INTEGRATION
BY USING MATRIX INVERSION
Dagmara Matlak, Jarosław Matlak, Damian Słota, Roman Wituła
Institute of Mathematics, Silesian University of Technology, Gliwice, Poland
[email protected]
Abstract. In the paper certain examples of applications of the matrix inverses for generating and calculating the integrals are presented.
Keywords: matrix inverse, integrals, generalized McLaurin’s formula
Introduction
The first part of our discussion concerns the linear mappings defined on the
finite-dimensional space of solutions of the following system of differential equations
= , + ⋯ + , = , + ⋯ + , ⋮
=
+
⋯ + , .
, (1)
Suppose that functions , , … , form the solution of the above system of
equations and matrix = , × of system (1) is nonsingular. Let us consider
the linear mapping of the linear space of solutions , , … , of system (1)
onto itself defined in the following way:
,
, + ⋯ + , ⋮
⋮ = ⋮ = = ⋮
,
, + ⋯ + , …
⋮
…
, ⋮ ⋮ = ⋮ . (2)
, Matrix is nonsingular, so there exists its inverse . In particular, the following equality occurs:
⋮
= ⋮ .
(3)
64
D. Matlak, J. Matlak, D. Słota, R. Wituła
Therefore, if ⋮
= ⋮ , then functions are the primitive functions
of ( = 1,2, … , ).
The current article is inspired by Swartz’s paper [1] where the author gives
some simple examples of using this procedure, among others, for generating the
integrals of functions ℎ = e
sin , ℎ = cos , for which he obtained the
formula (the integration constants are omitted and this rule will oblige henceforward):
ℎ =
ℎ ೌೣ
# − $%# − ℎ
! = మ మ "
&.
# + $%# ℎ
(4)
1. Generalization of Swartz’s example
Let us start from the generalization of the example mentioned above. Let us take
the functions
= cosh sin , = cosh cos ,
= sinh cos , = sinh sin .
(5)
Note that the differentiation operator for these functions is of the form
0
−
' ( = * , = 0
) +
0
0
0
0
0
. ' (.
− 0
(6)
If ≠ 0, then the inverse of matrix of operator has the form
=
మ మ 0
0
−
0
0
Now we can easily integrate functions , e.g.
/ =
0
0
−
0
..
0
− + .
+ (7)
65
Differentiation and integration by using matrix inversion
2. Integrals of ‫ ࢞ ࢔ܖܑܛ‬for odd ࢔
Consider the second derivatives of functions = sin , ≥ 2. We have
sin = sin cos = − 1 sin − sin .
(8)
sin − sin sin sin
6 sin − 9 sin sin ,='
'
(=
(=*
⋮
⋮
⋮
00 − 1 sin − 0 sin sin )1sin 2 +
−1
0
⋯
0
0
sin 0
0
3 ∙ 2 −3 ⋯
sin =.'
(.
⋮
⋮
⋱
⋮
⋮
⋮
0
0
⋯ 00 − 1 −0 sin (9)
Of course for = 1 there is sin = −sin . Thus we can write the second
derivative operator for the odd powers of function sin , from 1 to odd 0, in the
following matrix form:
The determinant of the obtained matrix is equal to: det = (−1)
0 ∈ 3. The inverse matrix is of the form
= , ೖశభ×ೖశభ ,
మ
where
, =
ೖశభ
మ
(0‼) ≠ 0,
(10)
మ
0,
− మ ,
‼ ‼
− ‼ ‼ ,
< 4,
= 4,
> 4.
(11)
We can deduce that for odd there occurs (with respect to the linear element):
sin = −
‼
‼
⁄
∑
()‼
sin ,
()‼
where : = . For example, we get
sin = − sin − sin − sin .
(12)
(13)
We note that from (12) by differentiating we obtain (see [2, 3]):
sin = −
‼
‼
cos ∑
⁄ ()‼
sin .
()‼
(14)
66
D. Matlak, J. Matlak, D. Słota, R. Wituła
For example, we have
sin = − cos (1 + sin + sin ).
3. The case of the even powers of ‫࢞ ܖܑܛ‬
sin ,
= sin − ೙షభ
೙
Consider functions of the form
(15)
for = 2,4,6, … . Acting on the vector '
(, where 0 is even, with the second
⋮
derivative operator, like it was done in equation (9), we get the following transformation matrix:
6 ' ( = ' (=
⋮
⋮
(16)
−2
0
…
0
0
2 ∙ 3⁄4 −4 …
0
0
=.' (.
⋮
⋮
⋱
⋮
⋮
⋮
0
0
… 0 − 2 0 − 1⁄0 −0 The above matrix is invertible and its inversion is of the form
, =
where
6 = , ೖ×ೖ,
మ మ
0,
− మ ,
‼ ‼
− మ ‼ ‼ ,
(17)
< 4,
= 4,
> 4.
(18)
Therefore for even n we get the formula (exact to the linear element):
= −
=
⁄
∑
"sin −
‼
‼
sin &
‼
"
sin − 1 +
‼మ ‼
‼
‼ ⁄
+ ∑ sin ‼ − ‼ &
= −
‼
‼మ
‼
‼
‼ మ
− మ sin = − మ sin ,
(19)
Differentiation and integration by using matrix inversion
67
which implies the following integral identity
sin =
=
‼
‼ మ
"
‼
‼
" + ∑
&
()‼
− ∑
‼
⁄
()‼()
Hence, by differentiating we get (see [2, 3]):
sin =
‼
‼
For example, we obtain
‼
⁄
& .
మ sin
" − cos ∑
⁄ ‼
sin &.
()‼
sin = − cos "sin + sin + sin &!.
(20)
(21)
(22)
4. Integral of ‫࢞ ࢔ܖ܉ܜ‬
Let 8 be the linear space of sequences 9 :
of differentiable functions
: (, ) → ;. Let <: 8 → 8 be a linear operator satisfying equation
tan 1
tan tan '
( = < ' (.
tan tan ⋮
⋮
(23)
If <is represented by infinite matrix , then from (23) matrix has the form
1 0 1 0 0 ⋮
0 2 0 2 0 ⋮
=..
0 0 3 0 3 ⋮
⋯ ⋯ ⋯ ⋯ ⋯ ⋯
It is easy to show that
0
⋮B
?1 0 − 0
>
0 − 0
⋮ A.
= >0 A
>0 0
0 − ⋮ A
=⋯ ⋯ ⋯
⋯ ⋯ ⋯@
(24)
(25)
However, matrix does not represent the inverse operator < , since the following relations hold
68
tan 1
0
tan (='
0
tan ⋯
⋮
resulting from the formula
D. Matlak, J. Matlak, D. Słota, R. Wituła
1
0 0 0 −1 0
0
0 ⋮
1 0 0
0 −1 0
0 ⋮
tan . ' (, (26)
0 1 0
0
0 −1 0 ⋮
tan ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
⋮
"
tan −
tan &
= tan − tan .
(27)
Moreover, we get from this, after summing over powers in the interval − , and
by uniform convergence (see [4]), that
− tan &,
tan = ∑
" tan
or equivalently
tan CC
= ∑
"
tan ೖ
= ∑
tan
,
−
tan &
(28)
(29)
for every ∈ − , and = 0,1,2, … .
From formula (28) we also get the matrix form D< of operator < , i.e. the
inverse operator of operator < (under assumption of its existence):
1
0
0 0
?
0
0
0
>
>− 0
0
D< = >
>0 −
0 > 0 − 0
> = ⋮
⋮
⋮
⋮
Formula (29) is our main analytic result in this
Because, as we show now, this formula is a
MacLaurin’s formulae for ln( + 1), i.e.
ln + 1 = −
మ
+
య
−
0 ⋯
B
0 ⋯A
0 ⋯A
A.
(30)
0 ⋯A
A
⋯
A
⋮ ⋱@
section. Why do we think so?
generalization of the classical
ర
+ ⋯,
(31)
for −1 < ≤ 1, found independently by Nicolaus Mercator and Saint-Vincent
(see sections 10-9 and 10-10 in [5] and page 387 in [6]), and for arctan , i.e.
arctan = −
య
+
ఱ
− ⋯,
for −1 < ≤ 1,which is known as the Gregory series.
(32)
69
Differentiation and integration by using matrix inversion
This connection should not be surprising because of the known complex relation (see [7]):
arctan E = ln "
&,
(33)
(−∞, −1] ∪ [1, ∞) and where the principal branch of the logawhere E⁄ ∈
rithm is under consideration. On the cuts we have
arctan(C) = ± + ln "
&,
(34)
for C ∈ (−∞, −1) ∪ (1, ∞) and where the upper/lower sign corresponds to the
right/left side of the set determining C. More precisely, the connection between the
arctan function and log function is obvious and the section concerns the real and
imaginary parts of arctan E, since we have
arctan E = arctan "
మ మ & + ln "
మ మ &,
(35)
()
− ln cos = ∑
tan
(36)
మ మ
where E = + C, |E| < 1. First, from equation (29) for = 1 we get
ೖ
or
ln cos1arctan √2 = ∑
ೖ
,
which by (31) implies the well known identity (since cos H =
H ∈ "− , &):
i.e.
(37)
√ !"మ #
ln cos1arctan √2 = − ln + 1,
(38)
√ + 1 cosarctan ≡ 1,
for every ∈ I0,1J.
But this formula holds for every ≥ 0 since cos H =
"− , &.
H∈
(29) we get
for
(39)
for every
In other words, formula (37) is equivalent to (31). For = 0 from
!$% !" = ∑
&
మ '
√ !"మ #
ೖ
,
(40)
70
D. Matlak, J. Matlak, D. Słota, R. Wituła
which implies (32). Hence, for = 1 we obtain
(
=1− + − +⋯
(41)
That is the classical Gregory-Leibniz-Nilakantha’s formula (see [8]). Generalizations of the Gregory power series (32) are discussed in papers [9] and [10].
As seen from equation (29), the values of integrals ర tan for ≥ 2 are
ഏ
the translations of numbers ర tan or ర , depending on parity of . For
even we have
ഏ
ഏ
ర tan = ర − ∑
ഏ
ഏ
whereas for odd we get
()⁄ ()ೖ
ర tan = ర tan − ∑
ഏ
ഏ
()⁄ ()ೖ
= − ∑
()⁄ ()ೖ
,
(42)
= "ln 2 − ∑
()⁄ ()ೖ
&.(43)
5. Final remark
Some other applications of the matrix obtained by n-times differentiation of
product functions and composition functions are discussed in paper [11]. In turn, in
paper [12] the technique of the inverse matrix was used for calculating the integral
sec , similarly as in the present study. The obtained formulae were used
there for generating the trigonometric identities.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
Swartz W., Integration by matrix inversion, Amer. Math. Monthly 1958, 65, 282-283.
Prudnikov A.P., Bryczkov J.A., Mariczev O.J., Integrals and Series, Elementary Functions,
Vol. 1, Nauka Press, Moscov 1986 (in Russian).
Prudnikov A.P., Bryczkov J.A., Mariczev O.J., Integrals and Series, Complements Sections,
Vol. 2, Nauka Press, Moscow 1986 (in Russian).
Kołodziej W., Mathematical Analysis, PWN, Warsaw 1979 (in Polish).
Eves H., An Introduction to the History of Mathematics, Holt, Rinehart and Winston, New York
1969.
Boyer C. (revised by Merzbach U.), A History of Mathematics, Wiley, New York 1991.
Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W., NIST Handbook of Mathematical Functions, Cambridge Univ. Press, Cambridge 2010.
Wituła R., Number π, its History and Influence on Mathematical Creativeness, Wyd. Pracowni
Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).
Differentiation and integration by using matrix inversion
[9]
71
Gawrońska N., Słota D., Wituła R., Zielonka A., Some generalizations of Gregory’s power
series and their applications, J. Appl. Math. Comput. Mech. 2013, 12(3), 79-91.
[10] Wituła R., Hetmaniok E., Pleszczyński M., Słota D., Generalized Gregory’s series (in review).
[11] Redheffer R., Induced transformations of the derivative - vector, Amer. Math. Monthly 1976,
83, 255-259.
[12] Wituła R., Matlak D., Matlak J., Słota D., Use of matrices in evaluation of ‫ ׬‬sec ଶ௡ାଵ ‫ݔ݀ ݔ‬
in review.

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