Full Text
UDC 517.512.6
MATHEMATICS
N. S. BAIGUZOV
APPROXIMATE DIFFERENTIATION BY MEANS OF LAGRANGE AND HERMITE INTERPOLATION POLYNOMIALS
(Presented by Academician M. A. Lavrent’ev, 15 I 1968)
As is well known, one of the methods for the approximate computation of a derivative is to replace it by the differentiated Lagrange interpolation polynomial (¹–³). In connection with this, for a function \(f(x)\) continuously differentiable on the segment \([-1,1]\), consider the quantity
\[ R_n'(f,x,M)=f'(x)-L_n'(f,x,M), \tag{1} \]
where
\[ L_n(f,x,M)=\sum_0^n f\bigl(x_k^{(n+1)}\bigr)\,l_{k,n+1}(x), \tag{2} \]
\[ l_{k,n+1}(x)= \frac{\omega_{n+1}(x)} {\omega_{n+1}'\bigl(x_k^{(n+1)}\bigr)\bigl(x-x_k^{(n+1)}\bigr)}, \qquad \omega_{n+1}(x)=\prod_0^n \bigl(x-x_k^{(n+1)}\bigr), \tag{3} \]
and \(M=\|x_k^{(n+1)}\|\) is a certain infinite triangular matrix of nodes located on the segment \([-1,1]\). In papers (¹, ⁴, ⁶) cases are considered in which the quantity (1) is estimated in terms of the derivative of order \(n\) of the function \(f(x)\).
In the present paper estimates are given for the rate of decrease of the quantity (1) under the condition that the function \(f(x)\) is only continuously differentiable on \([-1,1]\), and the matrix of nodes is defined by one of the formulas
\[ x_0^{(n+2)}=1,\qquad x_{n+1}^{(n+2)}=-1,\qquad x_k^{(n+2)}=\cos\frac{2k-1}{2n}\pi \quad (k=1,\ldots,n); \tag{4} \]
\[ x_k^{(n+1)}=\cos k\frac{\pi}{n} \quad (k=0,\ldots,n). \tag{5} \]
From (3) and (4) it follows that \(\omega_{n+2}(x)=(x^2-1)\widetilde T_n(x)\), where \(\widetilde T_n(x)\) is the Chebyshev polynomial of the first kind with leading coefficient equal to one. Similarly, under condition (5) we have \(\omega_{n+1}(x)=(x^2-1)\widetilde U_{n-1}(x)\), where \(\widetilde U_{n-1}(x)\) is the Chebyshev polynomial of the second kind.
Let us denote the matrices of nodes (4) and (5), respectively, by \(M_1\) and \(M_2\), and put
\[ D_n(x,M_1)=\sum_1^n |l_{k,n+2}'(x)|\sin\theta_k^{(n+2)}, \tag{6} \]
\[ D_n(M_1)=\max_{x\in[-1,1]}D_n(x,M_1). \tag{7} \]
Lemma 1. If the matrix of nodes is defined by formula (4), then the inequalities hold
\[ \frac{2}{\pi}\,n\ln\frac{1}{\sin\pi/n} < D_n(M_1) < \frac{26}{\pi}\,n\ln n+O(n), \]
\[ c_1 n \leq \max_{x\in\Delta_n} D_n(x,M_1)\leq c_2 n, \]
where the set \(\Delta_n\) consists of the points \(x_k^{(n)}=\cos k\pi/n\) \((k=1,\ldots,n-1)\) and the zeros of the derivative of the polynomial \(\omega_{n+2}(x)=(x^2-1)\widetilde T_n(x)\).
The proofs of these inequalities are obtained with the aid of formulas (2) and (3), if one takes into account that, by virtue of (4), here we have \(\omega_{n+2}(x)=(x^2-1)\pi_n(x)\). Analogous results also hold for the matrix \(M_2\).
Theorem 1. If the function \(f(x)\) is continuously differentiable on the segment \([-1,1]\) and \(\omega(\delta,f')\) is the modulus of continuity of its derivative \(f'(x)\), then for the Lagrange interpolation polynomial corresponding to the matrix \(M_1\) the inequalities
\[
\left|R'_{n+1}(f,x,M_1)\right|\leq c_3\omega(1/n,f')\ln n,\qquad x\in[-1,1],
\tag{8}
\]
\[
\left|R'_{n+1}(f,x,M_1)\right|\leq c_4\omega(1/n,f'),\qquad x\in\Delta_n
\tag{9}
\]
hold.
Under the conditions of Theorem 1, in paper \((^7)\) the existence is proved of polynomials \(\{Q_n(x)\}\) for which, for \(k=0,1\), the estimates
\[
\left|f^{(k)}(x)-Q_n^{(k)}(x)\right|\leq c_5
\frac{\sqrt{1-x^2}^{\,1-k}}{n^{1-k}}\,
\omega\left(\frac{\sqrt{1-x^2}}{n},f'\right),
\qquad x\in[-1,1].
\]
Using these polynomials in the inequality
\[
\left|f'(x)-L'_{n+1}(f,x,M_1)\right|
\leq
\left|f'(x)-Q'_n(x)\right|
+
\sum_1^n
\left|f\!\left(x_k^{(n+2)}\right)-
Q_n\!\left(x_k^{(n+2)}\right)\right|
\left|l'_{k,n+2}(x)\right|,
\]
with the aid of Lemma 1 we obtain the estimates (8) and (9).
It is not hard to prove that estimate (9) is sharp in order. Indeed, if, for example, \(f_1(x)=-x^2(1-x^2)\) for \(-1\leq x<0\) and \(f_1(x)=x^2(1-x^2)\) for \(0\leq x\leq 1\), then we shall have \(f'_1(x)\in \operatorname{Lip}1\) and \(f'_1(0)=0\). Therefore, for \(n=2p\) we find
\[
L'_{n+1}(f_1,0,M_1)=-\frac{1}{n\cos \pi/2n}.
\]
If the function \(f(x)\) has \(p\) continuous derivatives on the segment \([-1,1]\), with the \(p\)-th derivative having modulus of continuity \(\omega(\delta,f^{(p)})\), then in the case of the matrix \(M_1\) the estimates
\[
\left|R'_{n+1}(f,x,M_1)\right|
\leq
c_6\frac{\ln n}{n^{p-1}}\,
\omega\left(\frac1n,f^{(p)}\right),
\qquad x\in[-1,1],
\tag{10}
\]
\[
\left|R'_{n+1}(f,x,M_1)\right|
\leq
c_7\frac1{n^{p-1}}\,
\omega\left(\frac1n,f^{(p)}\right),
\qquad x\in\Delta_n
\tag{11}
\]
are valid.
Analogous facts also hold for the matrix \(M_2\). Further, one may consider a matrix \(M\) for which the conditions
\[
0<c_8\leq
\max_{[x_k^{(n)},x_{k+1}^{(n)}]}|\omega_n(x)|
\Big/
\max_{[x_i^{(n)},x_{i+1}^{(n)}]}|\omega_n(x)|
\leq c_9<\infty,
\]
are fulfilled, where \(\omega_n(x)\) is defined by formula (3) for the matrix \(M\), and \(x_0^{(n)}=1\), \(x_{n+1}^{(n)}=-1\). The convergence of the interpolation process for such matrices is considered in papers \((^{8,9})\). It is not hard to show that estimates of the form (8), (10) also hold here.
Let us now consider the quantity
\[
r_{2n+1}(f,x,M)=f'(x)-H'_{2n+1}(f,x,M),
\tag{12}
\]
where
\[
H_{2n+1}(f,x,M)=
\sum_0^n f\!\left(x_k^{(n+1)}\right)U_{k,n+1}(x)
+
\sum_0^n f'\!\left(x_k^{(n+1)}\right)V_{k,n+1}(x),
\tag{13}
\]
\[
U_{k,n+1}(x)=
\left[
1-
\frac{\omega''_{n+1}\!\left(x_k^{(n+1)}\right)}
{\omega'_{n+1}\!\left(x_k^{(n+1)}\right)}
\left(x-x_k^{(n+1)}\right)
\right]
l^2_{k,n+1}(x),
\tag{14}
\]
\[
V_{k,n+1}(x)=
\left(x-x_k^{(n+1)}\right)l^2_{k,n+1}(x).
\tag{15}
\]
Similarly to (6) and (7), here we set
\[ B_{n+1}(f,x,\alpha)=\sum_{0}^{n}|f'_k(x)|\sin^\alpha\theta_k^{(n+1)},\qquad \alpha\geqslant 0, \tag{16} \]
\[ B_{n+1}(f,E,\alpha)=\max_{x\in E}B_{n+1}(f,x,\alpha),\qquad E=[-1,1]. \tag{17} \]
Lemma 2. If the matrix of nodes is determined by formula (4), then for \(x\in[-1,1]\) the inequality
\[ B_{n+1}(V,x,1)\leqslant \frac{2}{n}\,|\omega_{n+2}'(x)| \left\{\frac{4}{\pi}\,|\widetilde T_n'(x)|\ln n+O(1)\right\}+O(1). \]
holds.
Lemma 3. If the matrix of nodes is \(M_1\), then the inequalities
\[ c_{10}n\ln n\leqslant B_{n+1}(U,[-1,1],2)\leqslant c_{11}n\ln n, \]
\[ c_{12}n\leqslant B_{n+1}(U,\Delta_n)\leqslant c_{13}n. \]
hold.
Theorem 2. If the matrix of nodes is \(M_1\) and the function \(f(x)\) is twice continuously differentiable on the segment \([-1,1]\), then the estimates
\[ |r'_{2n+3}(f,x,M_1)|\leqslant c_{14}\frac{\ln n}{n}\,\omega\left(\frac{1}{n},f''\right),\qquad x\in[-1,1], \tag{18} \]
\[ |r'_{2n+3}(f,x,M_1)|\leqslant c_{15}\frac{1}{n}\,\omega\left(\frac{1}{n},f''\right),\qquad x\in\Delta_n. \tag{19} \]
hold.
Theorem 3. If the matrix of nodes is \(M_2\) and the function \(f(x)\) is continuously differentiable on the segment \([-1,1]\), then the inequalities
\[ |r'_{2n+1}(f,x,M_2)|\leqslant c_{16}\omega\left(\frac{1}{n},f'\right)\ln n,\qquad x\in[-1,1], \tag{20} \]
\[ |r'_{2n+1}(f,x,M_2)|\leqslant c_{17}\omega\left(\frac{1}{n},f'\right),\qquad x\in\Delta_n', \tag{21} \]
hold, where \(\Delta_n'\) is the set of zeros of the derivative of the polynomial
\(\omega_{n+1}(x)=(x^2-1)U_{n-1}(x)\).
As examples show, inequalities (19) and (21) are exact in order. If the function \(f(x)\) has \(p\) continuous derivatives, then the estimates (18)—(21) are modified analogously to (10)—(11).
One can consider analogous questions for the matrices \(M_1^*\) and \(M_2^*\), obtained respectively from the matrices \(M_1\) and \(M_2\) by deleting the end points \(x=\pm1\).
Theorem 4. If the matrix of nodes is \(M_1^*\) and the function \(f(x)\) is twice continuously differentiable on the segment \([-1,1]\), then the estimates
\[ |R'_{n-1}(f,x,M_1^*)|\leqslant c_{18}\omega\left(\frac{1}{n},f''\right)\ln n,\qquad x\in[-1,1], \]
\[ |R'_{n-1}(f,x,M_1^*)|\leqslant c_{19}\frac{1}{n}\, \omega\left(\frac{1}{n},f''\right),\qquad x\in\Delta_n'', \]
hold, where \(\Delta_n''\) is the set of zeros of the derivative of the polynomial
\(\omega_n(x)=\widetilde T_n(x)\).
Theorem 5. If the matrix of nodes is \(M_1^*\) and the function \(f(x)\) is continuously differentiable on the segment \([-1,1]\), then the estimates
\[ |R'_{n-1}(f,x,M_1^*)|\leqslant c_{20}\frac{1}{\sqrt{1-x^2}}\, \omega\left(\frac{1}{n},f'\right)\ln n,\qquad x\in(-1,1) \]
\[ |R'_{n-1}(f,x,M_1^*)|\leqslant c_{21}\omega\left(\frac{1}{n},f'\right),\qquad x\in\Delta_n''. \]
Estimates of the same form are also valid for the case of Hermite interpolation. For example, if \(f'(x)\in \mathrm{Lip}\,1\) and \(x\in\Delta_n''\), then the estimate
\[ |r'_{2n-1}(f,x,M_1)|\leqslant c_{22}/n,\qquad x\in\Delta_n'' \]
holds.
On the other hand, if one sets \(f_2(x)=-x^2/2\) for \(-1\le x<0\) and \(f_2(x)=x^2/2\) for \(0\le x\le 1\), then \(f'_2(x)\in \operatorname{Lip}1\), \(f'_2(0)=0\), and for \(n=2p\) we have
\[ H'_{2n-1}(f_2,0,M_1^*)=-\frac12\,\frac1{n^2\sin \pi/2n}. \]
In conclusion, we note some results on the approximate differentiation of periodic functions. Denote by \(m_1^*\) the matrix of nodes
\[ x_k^{(n)}=\frac{2n-2k+1}{2n}\pi \quad (k=1,\ldots,2n), \]
and let there correspond to it the trigonometric interpolation polynomial \(t_{n-1}(f,x,m_1^*)\).
Theorem 6. If an even periodic function \(f(x)\) is continuously differentiable \(p\) times, and \(E_n(f^{(p)})\) is the best approximation of \(f^{(p)}(x)\) by trigonometric polynomials of degree not exceeding \(n\), then the estimate
\[ \left|f^{(p)}(x)-t_{n-1}^{(p)}(f,x,m_1^*)\right|\le (c_{23}+c_{24}\ln n)\,E_n(f^{(p)}). \]
holds.
In particular, if \(p=1\), then the first estimate of Theorem 5 follows from this. Analogous results are obtained also in the case where the matrices \(M_1\) and \(M_2\) are transformed in the corresponding way.
In all the estimates given above, certain (not best possible) values of the constants have been computed.
I express my gratitude to my scientific adviser P. K. Suetin for suggesting the topic and for valuable advice in carrying out this work.
Ural State University
named after A. M. Gorky
Received
8 I 1968
REFERENCES CITED
- D. F. Stephenson, Theory of Interpolation, Moscow—Leningrad, 1935.
- V. E. Milne, Numerical Analysis, IL, 1953.
- Sh. E. Mikeladze, Numerical Methods of Mathematical Analysis, Moscow, 1953.
- M. L. Brodskii, UMN, 13, issue 6 (84), 73 (1958).
- A. Schönhage, Num. Math., 5, No. 11, 301 (1962).
- A. Schönhage, Math. Zs., 94, No. 2, 1 (1966).
- I. G. Golub and I. B. Yudin, Mathematical Notes, 1, No. 2, 163 (1967).
- P. Erdös, Ann. Math., 43, 1 (1942).
- D. L. Berman, DAN, 176, No. 2, 239 (1967).