Application of Linear Interpolation in Problems of Recording Distributed Parameters
Abstract Generated abstract
This paper examines the use of piecewise linear interpolation in discrete systems for recording distributed technological parameters. It derives the standard interpolation error for uniformly spaced sensors in terms of the maximum second derivative of the measured function, giving a formula for selecting sensor spacing and the required number of readings for a prescribed error. The paper also treats measurement readings and node positions as approximate quantities and applies propagation of errors to determine the irreducible error of the interpolated value. The resulting expressions show that this error is largest at sampling points, smallest at interval midpoints by a factor of the square root of two, and that node-position error depends on the local gradient between adjacent samples.
Full Text
APPLICATION OF LINEAR INTERPOLATION IN PROBLEMS OF RECORDING DISTRIBUTED PARAMETERS
V. F. DYADIK, I. E. NAATS
(Presented by the scientific seminar of the Computing Laboratory of TPI)
Discrete methods are most widely used in the recording of technological parameters because of their speed and the compact form in which current information is presented.
Discrete recording systems may include devices that form an interpolating function from discrete readings.
In the present article the basic questions of applying the principle of linear interpolation to the recording of curves of parameter distribution in the discrete method of measurement are considered.
The principle of linear interpolation consists in the fact that the readings of two adjacent sensors are joined by a straight-line segment and, over the field length \([a,b]\) as a whole, it is piecewise linear in character. The equation of the interpolating straight line for any pair of readings \((y_i, y_{i+1})\) is written in the form
\[ P_1(x)=y_i+(y_{i+1}-y_i)\frac{x-x_i}{x_{i+1}-x_i}, \tag{1} \]
where \(x_0, x_1, x_2, \ldots, x_i\) are the interpolation nodes \((i=0,1,2,\ldots)\), and \(n+1\) is the number of sensors.
The value of the parameter at any point between readings along the length of the unit is thus determined by expression (1). In this case the absolute error \(\delta\) is determined by the expression
\[ \delta \leq \left| f(x)-P_1(x) \right|. \tag{2} \]
Below we consider the case of linear interpolation with a uniform step \(h=x_{i+1}-x_i\) for any pair of points. The remainder term of interpolation formula (1) for the interpolation interval \((x_i,x_{i+1})\) has the form [1]:
\[ R_1=\frac{f''(\xi)}{2}(x-x_i)(x-x_{i+1}). \tag{3} \]
Making the substitution \(t=\dfrac{x-x_i}{h}\), \((0 \leq t \leq 1)\), we obtain
\[ (x-x_i)(x-x_{i+1})=h^2t(t-1). \]
The product \(t(t-1)\) assumes its greatest value in absolute magnitude at \(t=\dfrac{1}{2}\), equal to \(\dfrac{1}{4}\).
In (3) the point \(\xi\) belongs to the interval \((x_i, x_{i+1})\). Introduce the quantity
\[ M_2=\max |f''(\xi)| \]
\[ a\leq x\leq b. \]
Then
\[ R_1=\frac{M_2h^2}{8}, \tag{4} \]
and the quantity \(h\) is determined by the expression
\[ h\leq \sqrt{\frac{8\delta}{M_2}}. \tag{5} \]
Relation (5) makes it possible to determine the distance between two neighboring nodes under linear interpolation for a prescribed value of the interpolation error \(\delta\), and the necessary number of readings on the interpolation interval \([a,b]\)
\[ n+1=\frac{b-a}{h}+1. \tag{6} \]
Formula (4) determines the interpolation error under the condition that the readings \(y_0,y_1,y_2,\ldots,y_n\) and the interpolation nodes \(x_0,x_1,x_2,\ldots,x_n\) are known exactly. However, in solving problems of recording distributed parameters, these quantities, as a rule, are known only approximately.
The error arising in this case is called irreducible, since it cannot be decreased by increasing the number of nodes, unlike the interpolation error (method error) determined by formula (4).
The readings \(y_i\) are known with measurement error of the parameter values at the reading points, \(\sigma_1\), and the error in fixing the interpolation nodes is \(\sigma_2\). It is necessary to determine the irreducible error as a function of the values \(\sigma_1\) and \(\sigma_2\).
This problem may be regarded as the problem of determining the magnitude of the error in indirect measurements from the known errors of direct measurements [2].
According to the theorem on the composition of partial errors, for equation (1) we have
\[ \sigma^2(x)= \left(\frac{\partial P_1}{\partial y_0}\right)^2\sigma_1^2+ \left(\frac{\partial P_1}{\partial y_1}\right)^2\sigma_1^2+ \left(\frac{\partial P_1}{\partial x_0}\right)^2\sigma_2^2+ \left(\frac{\partial P_1}{\partial x_1}\right)^2\sigma_2^2. \tag{7} \]
Having calculated the values of the partial derivatives, we obtain:
\[ \sigma^2(x)= \left[ 1-\frac{2}{h}(x-x_0)+\frac{2}{h^2}(x-x_0)^2 \right]\sigma_1^2+ \]
\[ +\frac{(y_1-y_0)^2}{h^4} \left[(x-x_0-h)^2+(x_0-x)^2\right]\sigma_2^2. \tag{8} \]
The first term of expression (8) is the component of the irreducible error due to the inaccuracy of the readings; the second term is due to the inaccuracy in fixing the interpolation nodes. Expression (8) determines the value of the irreducible error for any \(x\) in the interval between two readings. To determine the range of variation of \(\sigma\) on the interpolation interval, the expressions in the square brackets of formula (8) are examined for extrema,
and the values of $\delta(x)$ at the sampling points are determined. Accordingly, for the nodes $(x_0, x_1)$ we find
\[ \sigma(x_0)=\sqrt{\sigma_1^2+\frac{(y_1-y_0)^2}{h^2}\sigma_2^2}, \]
\[ \sigma(x_1)=\sqrt{\sigma_1^2+\frac{(y_1-y_0)^2}{h^2}\sigma_2^2}, \tag{9} \]
\[ \sigma\left(x_0+\frac{h}{2}\right) = \sqrt{ \frac{1}{2} \left[ \sigma_1^2+\frac{(y_1-y_0)^2}{h^2}\sigma_2^2 \right] }. \]
The maximum unavoidable error occurs at the sampling points; the minimum, smaller by a factor of $\sqrt{2}$, occurs at the midpoint of the interval $(x_0, x_1)$.
If $\sigma_2=0$, then expressions (9), respectively, take the form
\[ \sigma_y(x_0)=\sigma_1;\quad \sigma_y(x_1)=\sigma_1;\quad \sigma_y\left(x_0+\frac{h}{2}\right)=\frac{\sigma_1}{\sqrt{2}}, \tag{10} \]
i.e., the maximum unavoidable error of equation (1), due to the inaccuracy of measuring the samples, is equal to the measurement error itself.
The minimum is $\sqrt{2}$ times smaller than the measurement error. When $\sigma_1$ is equal to zero, there is an unavoidable error caused by the inaccuracy of fixing the interpolation nodes, under the condition that the samples $y_i$ themselves are specified exactly. The corresponding values of this error are determined by:
\[ \sigma_x(x_0)=\frac{y_1-y_0}{h}\sigma_2, \]
\[ \sigma_x(x_1)=\frac{y_1-y_0}{h}\sigma_2, \tag{11} \]
\[ \sigma_x\left(x_0+\frac{h}{2}\right)=\frac{y_1-y_0}{\sqrt{2}\,h}\sigma_2. \]
Analysis of expressions (11) shows that the unavoidable error caused by the inaccuracy of fixing the interpolation nodes depends on the quantity $\dfrac{y_1-y_0}{h}$—the gradient of the interpolated function between two neighboring samples. The maximum value of this error is likewise at the sampling points, and the minimum is at the midpoint of the interval.
Conclusions
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The principal issues in applying the principle of linear interpolation for recording curves of the distribution of a parameter in the discrete method of measurement have been considered.
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A method is given for calculating the necessary number of samples.
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The unavoidable interpolation error caused by the inaccuracy of the samples and by the inaccuracy of fixing the interpolation nodes has been determined.
LITERATURE
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I. S. Berezin, N. P. Zhidkov. Methods of Computation. Vol. 1, Fizmatgiz, Moscow, 1962.
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M. F. Malikov. Fundamentals of Metrology, Part 1. Mashgiz, Moscow, 1949.