JPH05266063A - Device for interpolating n-dimensional data - Google Patents

Abstract

PURPOSE:To interpolate a function value by effectively utilizing known data in respect to a little pair of data more than n+1 pieces without always having regularity. CONSTITUTION:At the dimensional data interpolating device to calculate the functional value by interpolation from a known data point in respect to a data dot sequence having one dependent parameter (functional value) corresponding to (n) pieces of independent parameters, the coordinate of a point to be interpolated is inputted from an inputter 1 to a function computing element 2, and (m) pairs of known data stored in a storage device 3 are inputted to the function computing element 2. At the function computing element 2, the (m) pairs of data are rearranged in the order close to the point to be interpolated, afterwards, the (n+1) pairs of first-order independent data are selected out of the (m) pairs of rearranged data, and the function value is calculated corresponding to the parameter value to be interpolated. After this interpolation processing, the function value is outputted from an output unit 4.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention is such that the function form of one dependent parameter with respect to n independent parameters is unknown or difficult to express, and n-dimensional data of n-dimensional data when a function value is obtained by interpolation from known data points It relates to an interpolator.

[0002]

2. Description of the Related Art Conventionally, when performing n-dimensional interpolation, a known data array is regularly given in an n-dimensional space, and a given function value is obtained by utilizing this regularity. For example, regarding the interpolation method when data is given in a grid array,
The following method (a) or (b) is usually adopted. In this case, for n-dimensional interpolation, at least 2 ⁿ A set of data is needed.

(A). 2 ⁿ surrounding the point to be interpolated A set of data sets (n-dimensional space) is searched, the subspace passing through the interpolation point for each parameter and the function value for the subspace are sequentially obtained, and the dimension is reduced to finally obtain a given function value.
In this method, 2 ⁿ surrounding the point to be interpolated The regularity of the data array is used to find the data set (n-dimensional space) of the set. (B). In the case of three dimensions or less, interpolation using a spline function may be performed, which also utilizes the regularity of the data array.

[0004]

SUMMARY OF THE INVENTION In the above conventional method, 1
At least 2 ⁿ to compute the function values Individual data sets were needed. In addition, even when a local data set is added to improve the accuracy of function calculation, regularity of the data is required, so at least n sets of data are required.

However, since this data set is usually created by experiments or simulations, it takes a lot of time and money. Furthermore, it may be difficult to obtain the function value at the required parameter position.

The present invention has been made in view of the above situation, and interpolates a function value by effectively utilizing known data as much as possible for a set of n + 1 or more linearly independent data which does not necessarily have regularity. It is an object of the present invention to provide an n-dimensional data interpolating device capable of saving labor in data creation by including an arithmetic unit for performing the operation.

[0007]

According to the present invention, for a series of data points having one dependent parameter (function value) for n independent parameters, a function value is obtained by interpolation from known data points. In an interpolating device for dimensional data, an input device for inputting coordinates of points to be interpolated, a storage device for storing m sets of known data, and rearrangement of m sets of data in the order close to the points to be interpolated, From the m sets of data after rearrangement, n + 1 sets of primary independent data are selected, and n + 1
It is characterized by comprising a function calculator for obtaining a function value for a parameter value to be interpolated from a hyperplane equation determined by a set of data, and an output device for outputting a function value after interpolation processing.

[0008]

First, the coordinates P [x ^{T of the} point to be interpolated by the input device ，
ζ] and input m sets of known data Pi from the storage device.
[X ^{(i) T} , ζ ⁽ⁱ⁾ ]. In the calculation unit, m
Each data point Pi [x ^{(i) T} , ζ ^{(i) of the set} ], The data point P [x ^T to be interpolated , Ζ ⁽ⁱ⁾ ], And the m sets of data after the rearrangement [x ^T , Ζ ⁽ⁱ⁾ ] First-order independent n + 1 sets of data are selected from among the above], a hyperplane equation determined by the selected n sets of data is created, and a parameter to be interpolated is substituted to obtain a function value. Then, the obtained function value is output by the output device.

[0009]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS An embodiment of the present invention will be described below with reference to the drawings. FIG. 1 is a block diagram showing the arrangement of an n-dimensional data interpolation apparatus according to an embodiment of the present invention. In FIG. 1, reference numeral 1 denotes an input device, which is a value of an independent parameter for obtaining a function value, that is, coordinates P [x ^T , Ζ] is input to the function calculator 2. Further, m sets (n + 1 ≦) in which one set of dependent parameters (function values) for n independent parameters input from the input device 1 is set as data.
The data collection of m) is stored in the storage device 3. The data stored in the storage device 3, that is, m sets of known data Pi [x ^{(i) T} , ζ ⁽ⁱ⁾ ] Is input to the function calculator 2. The function calculator 2 determines m sets of data points Pi.
[X ^{(i) T} , ζ ⁽ⁱ⁾ ], The data point P [x ^T to be interpolated ，
ζ ⁽ⁱ⁾ ], And the m sets of data after the rearrangement [x ^T , Ζ ⁽ⁱ⁾ ] First-order independent n + 1 sets of data are selected from among the above], a hyperplane equation determined by the selected n sets of data is created, and a parameter to be interpolated is substituted to obtain a function value. Then, the obtained function value is output by the output device 4.

The function calculator 2 is composed of a first calculator 2a, a second calculator 2b, and a third calculator 2c, as shown in FIG. The first calculation unit 2a rearranges the m sets of data in the order closer to the point to be interpolated. The second operation unit 2b selects n + 1 sets of data that are linearly independent. Third computing unit 2c
Performs calculation for obtaining a function value for the parameter value to be interpolated.

Next, the operation of the above embodiment will be described with reference to the flowchart of FIG. From the input device 1, the independent parameter value P [x ^{T of the} function to be interpolated , Ζ] is a function calculator 2
(Step A1). In addition, from the storage device 3, a set of independent parameters and their function values [x ^{(i) T} , ζ
⁽ⁱ⁾ ] Are read out and input to the function calculator 2 (step A2).

In the function calculator 2, first, in the first calculator 2a, m sets of data points Pi [x ^{(i) T} , ζ ^{(i) are set.} ]When,
Data point P [x ^T to be interpolated , Ζ] in the n-dimensional space is obtained, and the arrangement of the points Pi is changed in ascending order of di (step A3). Next, the second computing unit 2b causes the m sets of data [x ^T after rearrangement] to be changed. , Ζ ⁽ⁱ⁾ ], The first independent n + 1 sets of data are selected (step A4).
After that, the third computing unit 2c creates a hyperplane equation determined by n + 1 sets of data, and substitutes the parameter value to be interpolated to obtain the function value (step A5). Then, the function value obtained by the function calculator 2 is output by the output device 4 (step A6). Next, the first arithmetic unit 2a, the second arithmetic unit 2b, and the third arithmetic unit 2 in the function arithmetic unit 2 described above.
The specific processing of c will be described. [Processing of the first calculation unit 2a]

The independent parameters are x = [ξ1, ξ2, ... ξ
n] ^T , X is written as ζ (scalar). Preliminarily m sets of data [x ^{(i) T} , ζ ⁽ⁱ⁾ ]
It is assumed that (i = 1, 2, ..., M) are numbered so that they are arranged in ascending order from the given point x. That is,

[0014]

[Equation 1] [Processing of the second calculation unit 2b] By the Schmidt orthogonalization method,
(N + 1) sets of primary independent vectors are selected by the following procedure.

[0015]

[Equation 2][Processing of Third Computing Unit 2c] Here, the problem is that "n + 1 sets of
Point value [x^{(i) T}, Ζ⁽ⁱ⁾ ] (I = 1, 2, ... N + 1, point
No.) is given, ζ = f (x) where ζ⁽ⁱ⁾ = F (x)⁽ⁱ⁾  To obtain a function f that satisfies

The simplest function f is the n + 1-dimensional [x ^T which can pass n + 1 points. , Ζ] space is represented by a hyperplane. That is, n + 1 independent points [x ^{(i) T} , ζ ⁽ⁱ⁾ ] Is the hyperplane equation

[0017]

[Equation 3] This can be summarized as follows for i, j = 1, 2, ..., N + 1. A ・ B = E

[0018]

[Equation 4] Therefore, if A is non-singular, the simultaneous linear equation “A
B can be solved by solving "B = E" (B is equal to the inverse matrix A ^-1 of A). The necessary and sufficient condition for A to be non-singular is that the vector sequence [x ^{(i) T} , 1] (i = 1, 2,
.. (n + 1) are (n + 1) -space and are first-order independent of each other. The simplest two-dimensional (n = 2) interpolation will be specifically described below as an example. The problem is,

"Two (n) independent parameters x, y
On the other hand, when a data point sequence having one dependent parameter ζ is given, the dependent parameter value ζA at the independent parameter value (xA, yA) is obtained by interpolation. It is. Here, the point sequence is generated assuming the function form as follows. ζ = sin (x + y) where 0 ≦ x ≦ π / 2 0 ≦ y ≦ π / 2 Point A (xA, yA) = (1.7π / 10, 1.5π / 1
0)

The sequence of data points does not have to be grid-like
The point sequence shown in Fig. 4 is increased by increasing the points only in the part where the change is large.
And In FIG. 4, the sequence of points is (xi, yi, ζi)
(I = 1 to 16), and the score is “16 points”. Data points
Is added to any position as shown in Fig. 5 from a minimum of 1 point.
It is possible. The numbers in parentheses in FIG. 4 are point numbers.
Then, the processing is performed in the following procedure. (1). Arrange the given 16 points in the order close to point A
It di = (xi-xA)² ＋ (yi-y A)²  Therefore, as shown in FIG. 6, "d6 <d3 <d7, ..."
It

(2). 3 points that are independent in the first order (= n
Select +1 point). Since the points (6, 3, 7) shown in FIG. 6 are not in a straight line, they can be used as primary independent data. (3). Create a hyperplane equation that is determined by the three points (6, 3, 7). ζ = a1 x, a2 y, + a3

[0022]

[Equation 5] When a value is substituted into this simultaneous equation and solved, a1 = (S1−S3) 10 / 2π a2 = 2S2−S1 a3 = (S3−2S2 + S1) 10 / 2π where S1 = sin (4π / 10) S2 = Sin (2π / 10) S3 = sin (2π / 10) is obtained. That is,

[0023]

[Equation 6] From this, a1 = (S1-S3) 10 / 2.pi.a2 = 2S2-S1a3 = (S3-2S2 + S1) 10 / 2.pi. Is obtained. (4). Find the interpolated value ζAA. The interpolated value ζAA is obtained by ζA = a1 xA + a2 yA + a3. Specifically, ζA = (1.7π / 10) · (10 / 2π) (S3−2S2 + S1) + (1.5π / 10) · (10 / 2π) (S1−S3) + 2S2− S1 = (1.7 / 2) S3−1.7 S2 + (1.7 / 2) S1 + (1.5 / 2) S3 + 2 S2−S1 = (1.2 / 2) S1 + 0.3 S2 + (0.2 / 2) S3 = 0.805748 can get. Since the true value is 0.84432793, the error is 4.569%. The accuracy is improved by adding another point between the points (6) and (7).

When the above two-dimensional data is interpolated, the required number of data points X is "X ≧ n" as shown in FIG.
+ 1 ≧ 3 (n = 2) ”, and as compared with the conventional example shown in FIG. 8, a function value with the same accuracy can be obtained with fewer data points. In the conventional example of FIG. 8, the required number of data points X is “X ≧
2 ⁿ ≧ 4 (n = 2) ”, and in order to improve the accuracy of the function, it is necessary to add the data of two points a and b.

[0025]

As described above in detail, according to the present invention, it is not necessary to have regularity, and known data can be effectively used for a small data set of n + 1 or more to interpolate function values. Can be done.

[Brief description of drawings]

FIG. 1 is a block diagram showing a configuration of an n-dimensional data interpolation device according to an embodiment of the present invention.

FIG. 2 is a block diagram showing the configuration of a function calculation unit in FIG.

FIG. 3 is a timing chart explaining the operation of the embodiment.

FIG. 4 is a diagram showing a data point sequence during interpolation processing for two-dimensional data.

FIG. 5 is a diagram for explaining addition of a data point sequence.

FIG. 6 is a diagram showing a process of rearranging point data.

FIG. 7 is a diagram showing an interpolation operation of two-dimensional data in the present invention.

FIG. 8 is a diagram showing a conventional two-dimensional data interpolation operation.

[Explanation of symbols]

DESCRIPTION OF SYMBOLS 1 ... Input device, 2 ... Function operation device, 2a ... 1st operation part, 2b
... second operation unit, 2c ... third operation unit, 3 ... storage device, 4 ...
Output device.

Claims (1)

[Claims] 1. An interpolation device for n-dimensional data, which calculates a function value from a known data point by interpolation for a data point sequence having one dependent parameter (function value) for n independent parameters. An input device for inputting coordinates of a point to be stored, a storage device for storing m sets of known data,
After rearranging the m sets of data in the order closer to the point to be interpolated,
A function calculator that selects a primary independent n + 1 set of data from the rearranged m sets of data, and obtains a function value for a parameter value to be interpolated from a hyperplane equation determined by the n + 1 set of data; An interpolator for n-dimensional data, comprising: an output device that outputs a function value after the interpolation processing.