JPH06337775A - Two-dimensional numerical integral calculation device - Google Patents

Abstract

PURPOSE:To calculate two-dimensional numerical integral values of a function by the two-dimensional numeric integral calculation device irrelevantly to integra tion order. CONSTITUTION:This device is equipped with a 1st calculating means 11 which sets section division points in an internal section having values as many as coordinate points of an object of internal integration as section division points of external integration when the coordinate points are supplied and calculates internal integral values according a prescribed integrating method by using the function values at those section division points, a 2nd calculating means 14 which calculates external integral values according to a prescribed integrating method by using the internal integral values when the 1st calculating means 11 calculates the internal integral values, a judging means 15 which judges the convergence of the external integral values calculated by the 2nd calculating means 14, and an actuating means 16 which sets section division points of external integration having larger values than before when the judging means 15 judges nonconvergence and actuates the 1st calculating means 11.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a two-dimensional numerical integral calculating device for calculating a two-dimensional numerical integral value of a given function, and more particularly, to enable calculation of a two-dimensional numerical integral value regardless of the order of integration. And a two-dimensional numerical integration calculation device.

In various analysis systems, it is necessary to calculate a two-dimensional numerical integral value of a function. For example, in an electromagnetic wave analysis system, a mutual impedance matrix is calculated to predict an electromagnetic radiation amount and an electromagnetic wave shielding effect is analyzed. To calculate the mutual impedance matrix, a two-dimensional numerical integral value of a function is calculated. It is required to calculate.

The calculation of the two-dimensional numerical integral value of this function must be executed so that the same result can be obtained regardless of the order of integration.

[0004]

2. Description of the Related Art Conventionally, in the case of calculating a two-dimensional numerical integral value of a function, a configuration has been adopted in which the convergence is determined separately for the inner integral and the outer integral. Midpoint rule, trapezoidal rule, Gauss
Law, Simpson's law, IMT law, etc. are used, but conventionally, no matter which integration method is used to calculate a two-dimensional numerical integration value, the inner integration and the outer integration are separately performed. It adopted a configuration that determines the convergence.

That is, in the prior art, first, the number of section divisions of the outer integral is set to a certain value, the convergence control of the inner integral is performed while increasing the number of section divisions of the corresponding inner integral, and the converged inner integral value. If the outer integral is not converged, the same calculation should be repeated while increasing the number of divisions of the outer integral until the outer integral converges. Then, the configuration of calculating the two-dimensional numerical integral value is adopted.

To explain more specifically,

[0007]

[Equation 1]

When obtaining the two-dimensional numerical integral value of the function f represented by

[0009]

[Equation 2]

In this way, the variable y is first integrated,
After that, a method of integrating the variable x (hereinafter sometimes referred to as the former integration method),

[0011]

[Equation 3]

In this way, the variable x is first integrated,
After that, there is a method of integrating the variable y (hereinafter sometimes referred to as the latter integration method). In this case, if the latter integration method is explained, first, as shown in FIG. 11, at each interval division point y _i

[0013]

[Equation 4]

The value of [Equation 3] is calculated using this value, and in the prior art,
The number of interval divisions for the variable x that realizes the convergence of the value of the expression (4) (there may be a different value for each variable y _i ) is the variable that realizes the convergence of the value of the expression (3). The method of setting is adopted regardless of the number of divisions of y (that is, the number of y _i ).

Next, the conventional technique based on the Simpson rule will be described in detail. In the conventional technique based on Simpson's law, when following the former integration method, specifically,

[0016]

[Equation 5]

The integral value is calculated according to the above. Here, “T _y (x _i )” in the equation is

[0018]

[Equation 6]

Where m is the number of section divisions when the outer integral converges, and n _i (depending on x _i ) is the number of section divisions when the inner integral converges,
“X _i ” and “y _j ” in the formula are

[0020]

[Equation 7]

Is defined by Also, Simpson
In the conventional law-based technology, if the latter integration method is followed, specifically,

[0022]

[Equation 8]

The integrated value is calculated according to the following. Here, “T _x (y _i )” in the equation is

[0024]

[Equation 9]

[Mathematical formula-see original document] where m is the number of section divisions when the outer integral converges, and n _i (depending on y _i ) is the number of section divisions when the inner integral converges,
“X _j ” and “y _i ” in the equation are

[0026]

[Equation 10]

Is defined by 12 and 1
FIG. 3 illustrates a processing flow of a conventional technique based on this Simpson rule.

In the prior art, first, in "BOX1" of the processing flow of FIG. 12, the subroutine shown in the processing flow of FIG. 13 is called by using the center point and the end points of the outer section as parameters. The inner integral values for these three section dividing points are calculated.

Here, in the processing flow of FIG.
In “BOX5”, a process of acquiring the function value f with respect to the center point and both end points of the inner section and calculating the inner integrated value T (corresponding to S ₀ of the equation (13) described later) is executed, and “BOX
In "X6", a process (specifically, 2n → n is set) of setting a new number of new segment division points in the inner segment in order to converge the inner integral value (specifically, 2n → n) is executed, and "judgment C" and " B
In “OX7”, the function value of the new section dividing point set in “BOX6” is acquired, and the partial sum dT of the function value f (the value of the functional sum part in S _{pk of the} formula (14) described later is obtained. (Corresponding) is executed, and in "BOX8", "BOX"
7 ”, a process of updating the inner integral value T is performed according to the partial sum dT of the function value f updated in step 7 and the previously calculated inner integral value and partial sum. By executing an error between the calculated inner integral value and the previously calculated inner integral value, it is determined whether or not the updated inner integral value has converged. When this "judgment D" results in a non-convergence judgment, "BOX6" is executed again, and in the case of Simpson's rule, a new section division point is added to the intermediate position of the previous section division points. Will be.

Then, the "BOX" in the processing flow of FIG.
2 ”, set a new specified number of additional section division points in the outer section in order to converge the outer integral value, and then,
By using the "judgment A" and "BOX3" with the new section dividing point as a parameter, the subroutine shown in the processing flow of FIG. 13 is called to calculate the inner integral value of the new section dividing point.

Subsequently, the "BOX4" is used to update the outer integral value obtained last time using the newly calculated inner integral value, and then the "outer integral value" is updated by "Judgment B". When it is determined that the error between the obtained outer integral value and the previously obtained outer integral value is within the specified range and it is determined that the error is within the error, it is determined that the updated outer integral value has converged, and the processing ends. . When it is determined that the difference is not within the error, the process returns to "BOX2" and a new additional segment division point is set in the outer segment to realize the convergence of the outer integral value.

Here, "BOX" in the processing flow of FIG.
4 ”and the“ BOX8 ”in the processing flow of FIG. 13, the update processing of the one-dimensional Simpson rule is used. To briefly explain this update process, 1
For the interval [a, b] of the function f by the dimensional Simpson rule

[0033]

[Equation 11]

Since it is known that the following equation holds, the relational expression is expressed by the sections [a, (a + b) / 2] and the section [(a +
b) / 2, b]

[0035]

[Equation 12]

It is possible to improve the accuracy of the integrated value of. If this is expressed by a mathematical expression, as an initial value,

[0037]

[Equation 13]

By defining

[0039]

[Equation 14]

Therefore, since the integral value S _{k of the} equation (14) is updated, "B" in the processing flow of FIG.
OX4 ”or“ BOX8 ”in the processing flow of FIG.
The update process is executed using this recurrence formula.

[0041]

As described above, conventionally, when the two-dimensional numerical integral value of the function is calculated, the convergence is determined separately for the inner integral and the outer integral.

However, according to such a conventional technique, there is a problem that the integration result differs depending on which of the two integration variables is allocated to the inside / outside. This is because when performing convergence determination separately for the inner integral and the outer integral, as shown in FIG. 14, when sampling the function value used when the former integration method is followed and when the latter integration method is followed. The reason is that the sampling point of the function value adopted in does not always match.

When this happens, the mutual impedance matrix obtained by the electromagnetic wave analysis system becomes an asymmetric matrix even though it is originally a symmetric matrix.
Various inconveniences will occur.

Further, according to such a conventional technique, there is a problem that the two convergence setting values, that is, the convergence setting value of the inner integral and the convergence setting value of the outer integral must be properly set in good balance. .

The present invention has been made in view of the above circumstances, and an object of the present invention is to provide a new two-dimensional numerical integral calculating device that can calculate a two-dimensional numerical integral value of a function regardless of the order of integration. To do.

[0046]

FIG. 1 shows the principle configuration of the present invention. In the figure, reference numeral 1 denotes a two-dimensional numerical integral calculation device equipped with the present invention, which calculates a two-dimensional numerical integral value of a given function.

This two-dimensional numerical integral calculation device 1 includes a function information management means 10, a first calculation means 11, an interval division point setting means 12, a storage means 13, and a second calculation means 14.
And a determination means 15, an activation means 16, and a section division point setting means 17.

The function information management means 10 manages function information to be integrated. The first calculating means 11 is for calculating the inner integral value of the two-dimensional numerical integration, and is provided with the section dividing point setting means 12 for setting the section dividing points of the inner section. The storage unit 13 stores the inner integrated value calculated by the first calculating unit 11 and the intermediate calculated value used for the calculation process of the inner integrated value while updating them.

The second calculating means 14 is the first calculating means 1.
The inner integral value calculated by 1 is used to calculate the outer integral value of the two-dimensional numerical integration. The determination means 15 is the second calculation means 1
The convergence of the outer integral value calculated in 4 is determined. Starting means 1
Reference numeral 6 is for activating the first calculating means 11, and is a section dividing point setting means 17 for setting the section dividing points of the outer section.
Equipped with.

[0050]

According to the present invention, when the section dividing point setting means 17 of the starting means 16 sets the section dividing points of the outer section according to the specified algorithm, the starting means 16 sets these section dividing points as the coordinate points of the inner integration target. After setting, the first calculation means 11 is activated.

When activated in this way, the section dividing point setting means 12 of the first calculating means 11 determines the section dividing points of the inner section having a value equal to the number of coordinate points designated by the starting means 16. , Is set for each coordinate point of each inner integration target according to a specified algorithm, and in response to this setting, the first calculation means 11 refers to the management data of the function information management means 10, and according to the specified integration method, Using the function values of these section dividing points, the inner integral value of the two-dimensional numerical integration is calculated for each coordinate point of the inner integral target.

When the inner integrated value is calculated in this way, the second calculating means 14 uses the calculated inner integrated value to form the outer integrated value which becomes a two-dimensional numerical integrated value, according to a prescribed integration method. The value is calculated, and in response to this calculation result, the judging means 15 judges whether or not the outer integral value has converged by comparing the calculated outer integral value with the previous calculated value.

When judging the convergence of the outer integral value by this judgment, the judging means 15 outputs the converged outer integral value to the outside as a two-dimensional numerical integral value, while when judging the non-convergence, the judgment is made. To the activation means 16. Then, upon receiving this notification, the activation means 16
A new coordinate point that is a target of inner integration showing a larger value than the previous time is set, and the first calculating means 11 is activated.

As described above, according to the present invention, when the two-dimensional numerical integral value of the function is calculated, the number of section division points of the inner integration is controlled to be equal to the number of section division points of the outer integration. By adopting a certain configuration, it becomes possible to calculate the two-dimensional numerical integral value regardless of the order of integration.

In the configuration of the present invention, when the trapezoidal rule, the Simpson's rule, or the IMT rule is used as the integration method, a new section dividing point is added to the previous section dividing point, and the section of the outer integration is added. The number of division points will be increased.
At this time, since the number of section divisions of the inner integral changes corresponding to the number of section divisions of the outer integral, the accuracy of the inner integral value is not high at the beginning of the calculation, but the accuracy is updated as the calculation progresses.

From this, the first calculating means 11 calculates the inner integrated value, and if there is the inner integrated value and the intermediate calculated value used in the calculation processing of the inner integrated value as in the Simpson rule, In addition, the intermediate calculation value and the intermediate calculation value are temporarily saved in the storage unit 13, and the starting unit 16 calculates the inner integration target coordinate point for which the inner integration value has already been calculated.
The corresponding inner integral value (inner integral value, intermediate calculated value) stored in the storing means 13 is used as notification information, and the first calculating means 11 is used.
When the internal calculating values (internal integrating value, intermediate calculated value) are notified from the starting means 16, the first calculating means 11 adopts a configuration for activating the internal calculating values (internal integrating value, intermediate calculating value). It is preferable to adopt a configuration in which the value is used to calculate a new inner integral value (inner integral value, intermediate calculation value) with higher accuracy. This is because the two-dimensional numerical integration value can be calculated at high speed by adopting this processing configuration.

[0057]

EXAMPLES The present invention will be described in detail below with reference to examples. 2 to 5 show an embodiment of a processing flow executed by the two-dimensional numerical integration calculation device 1 of the present invention when the Simpson rule is used as a base.

Next, the present invention will be described in detail according to this processing flow. Here, the process flow shown in FIGS. 2 to 4 executes a process of calculating a two-dimensional numerical integral value while calling the process flow shown in FIG. 5, while the process flow shown in FIG. When called from the process flow shown in FIG. 4, the process of calculating the inner integral value for the designated coordinate point of the inner integral target (which becomes the section dividing point of the outer integral) is executed.

First, the processing flow shown in FIG. 5 will be described. When the two-dimensional numerical integration calculation device 1 starts execution of the processing flow shown in FIG. 5, the “BOX7” is used to acquire the function value f for the center point and both end points of the inner section, and the inner integrated value T ( (Corresponding to S ₀ in the above formula [13])
To calculate. The inner integrated value T calculated at this time and the intermediate calculated value T _p (corresponding to S _P0 of the above-mentioned [Equation 13]) calculated from the function value of the center point of the inner section acquired at this time are The data will be saved in a specified data area area for use in updating the inner integral value. Note that this process is omitted except for the first time when the process is started.

Subsequently, it is judged by "judgment B" whether or not the number of section dividing points of the inner integral specified by the calling source is set, and when it is judged that they are not set,
By proceeding to "BOX8" and setting a new section dividing point between the section dividing points set up to that point, the number of section dividing points of the inner integration is doubled.

Subsequently, the "judgment C" and "BOX9" are used to obtain the function value of the new section dividing point set in "BOX8", and the partial sum dT of the function value f (the above [Equation 1]
4] Equivalent to the value of the function sum portion in S _{pk of the} equation) is updated.

When this update is completed, "BOX10" is displayed.
Going forward, the partial sum d of the function value f updated by "BOX9"
Intermediate calculation value T calculated from T_q(Equation 14 above formula
Of S _pkAnd the inner integrated value T calculated previously (the above-mentioned
S in the formula [14]_k-1Equivalent to) and the intermediate total calculated last time
Calculated value T_p(S in the above equation [14]_pk-1Equivalent to) and
Then, the inner integral value T is updated. Inner product calculated at this time
Minute value T, intermediate calculation value T _qIs the update process of the next inner integral value.
Save to a specified data area area for logical use.
Becomes

The process of updating the inner integral value is "judgment B".
Therefore, the process is executed until the setting of the interval dividing points of the inner integral specified by the calling source is confirmed. In this manner, the two-dimensional numerical integration calculation apparatus 1 executes the processing flow shown in FIG. 5 to set the number of inner division interval division points designated by the caller and set the inner integration value to the inner integration value. It processes so that it may be calculated.

Next, the processing flow of FIGS. 2 to 4 will be described. By executing this processing flow, the two-dimensional numerical integration calculation device 1 doubles the interval division points of the outer integration when the two-dimensional numerical integration values have not converged.
Then, the newly set section division points of the outer integration and the already set section division points of the outer integration are specified as the calculation targets of the inner integration, and the section division points of the outer integration are set as the points of the section division points of the inner integration. By designating the number of points and calling the subroutine shown in the processing flow of FIG. 5, the inner integral values for the interval division points of those outer integrals are acquired. Then, by calculating the outer integral value using the acquired inner integral value and determining whether or not the convergence has occurred,
The convergent two-dimensional numerical integral value is calculated. Here, when the subroutine shown in the processing flow of FIG. 5 is called, the inside integral value T and the intermediate calculation value T _p acquired at the time of the previous call are notified, so that this subroutine follows the updating process to obtain a new inside integral value. It will be processed so that the value can be calculated.

Specifically, the "BOX" in this processing flow is
1 ”to call the subroutine shown in the processing flow of FIG. 5 with the central point and both end points of the outer section as parameters to newly obtain the inner integrated values T for these three section dividing points. Also, the inner integration value T is updated by calling the subroutine shown in the processing flow of FIG. 5 in correspondence with the increase in the interval division points of the outer integration.

Also, with "BOX2", the interval division points of the outer integration are doubled. Also, with "BOX3",
Among the interval division points of the doubled outer integral, the inner integral value is updated by calling the subroutine shown in the processing flow of FIG. 5 for the inner integral value calculated last time. In addition, with "BOX4", "BOX3"
The outer integral value is updated according to the inner integral value updated in.

In addition, by calling the subroutine shown in the processing flow of FIG. 5 for “BOX5” among the section dividing points of the doubled outer integral, for which the inner integral value has not yet been calculated, Calculate the inner integral value. Further, the outer integral value is updated by "BOX6" according to the inner integral value calculated in "BOX5". Then, the "judgment A" is used to judge the convergence of the outer integral value.

The two-dimensional numerical integration processing of the present invention based on the Simpson rule will be described according to mathematical expressions. In the case of the former integration method,

[0069]

[Equation 15]

The integrated value is calculated in accordance with Here, “T _y (x _i )” in the equation is

[0071]

[Equation 16]

[Mathematical formula-see original document] where m is the number of interval divisions when the two-dimensional numerical integration converges, and "x _i " and "y _j " in the equation are:

[0073]

[Equation 17]

Is defined by Also, when following the latter integration method,

[0075]

[Equation 18]

The integrated value is calculated in accordance with Here, “T _x (y _i )” in the equation is

[0077]

[Formula 19]

[Mathematical formula-see original document] where m is the number of interval divisions when the two-dimensional numerical integration converges, and " _xj " and " _yi " in the equation are

[0079]

[Equation 20]

Is defined by Next, the processing of the processing flow of FIGS. 2 to 5 will be specifically described with reference to the processing explanatory diagram shown in FIG.

As shown in FIG. 6, at the first time, "i = 0, 4, 8" is set as the section dividing points of the outer integration, and correspondingly, the section dividing points of these outer integrations are set. On the other hand, the inner integration value is calculated by setting three inner integration interval dividing points “j = 0, 4, 8”, and the outer integration value is calculated using these inner integration values. It

Next, at the second time, "i = 2, 6" is added as the interval dividing point of the outer integration, in addition to "i = 0, 4, 8", and correspondingly, these For each interval dividing point of the outer integral, 5 called “j = 0, 2, 4, 6, 8”
The inner integral value is calculated by setting the interval dividing points of one inner integral, and the outer integral value is calculated using these inner integral values. At this time, since the line part consisting of the section dividing points of the black circles is the section dividing point of the inner integral defined by the new section dividing point of the outer integral, it is newly added without using the previous inner integral value. The inner integral value will be calculated. That is, it is necessary to newly obtain the function value for the black circle portion.

On the other hand, the row portion in which the section dividing points of the white circles and the section dividing points of the double circles are mixed is the section dividing point of the inner integral defined from the section dividing point of the outer integral set previously. Therefore, the inner integral value is calculated by using the inner integral value obtained based on the function values up to the previous time. That is, it is necessary to newly obtain the function value this time for the double circle portion, but it is not necessary to obtain the function value this time for the white circle portion.

Subsequently, at the third time, in addition to "i = 0, 2, 4, 6, 8" as the section dividing point of the outer integration, "i =
1, 3, 5, 7 ”is added, and correspondingly,“ j = 0, 1, 1, ”for each interval dividing point of these outer integrals.
The inner integral values are calculated by setting the nine inner integral interval dividing points "2, 3, 4, 5, 6, 7, 8", and the outer integral values are calculated using these inner integral values. It is calculated.

As described above, according to the present invention, when the two-dimensional numerical integral value of the function is calculated, the inner integral is not separated from the outer integral and converged, but the score of the interval dividing points of the inner integral is calculated. And the two-dimensional numerical integration value are converged while controlling so that the number of section division points of the outer integration becomes equal.

7 to 9 show an embodiment of a processing flow executed by the two-dimensional numerical integration calculating apparatus 1 of the present invention when the IMT rule is used as a base. Before entering the description of this processing flow, the one-dimensional IMT rule will be described.

In the middle point rule and the trapezoidal rule, the intervals between the interval dividing points are equal, whereas in the Gauss rule and the IMT rule, the intervals between the interval dividing points are unequal. The numerical integration value S of the equation (12) in the case where the intervals between the section dividing points are unequal intervals,

[0088]

[Equation 21]

It can be expressed as follows. Here, w _i is a weight and r _i represents an interval dividing point. And the IMT rule in this is as an initial value,

[0090]

[Equation 22]

When defined as

[0092]

[Equation 23]

It will be expressed by the following recurrence formula. Next, the processing flow of FIGS. 7 to 9 will be described. Here, the process flow shown in FIGS. 7 and 8 executes a process of calculating a two-dimensional numerical integral value while calling the process flow shown in FIG. 9, while the process flow shown in FIG. Also, when called from the process flow shown in FIG. 8, a process of calculating the inner integral value for the designated coordinate point of the inner integral target (which becomes the section dividing point of the outer integral) is executed.

First, the processing flow shown in FIG. 9 will be described. When the two-dimensional numerical integration calculation device 1 starts execution of the processing flow shown in FIG. 9, the “BOX7” is used to acquire the function value f with respect to the center point of the inner section and calculate the inner integration value T. At this time, the inner integrated value T is saved in the specified data area area for use in the next update processing of the inner integrated value. In addition, 1 of the process start
This process is omitted except for the first time.

Subsequently, it is judged by "judgment B" whether or not the number of inner division interval division points designated by the calling source is set, and when it is judged that they are not set,
By proceeding to "BOX8" and newly setting interval division points according to a prescribed algorithm, the number of interval division points of the inner integral is doubled.

Subsequently, the "judgment C" and "BOX9" are used to obtain the function value of the new section dividing point set in "BOX8", and the partial sum dT of the weighted function value f is obtained.
(Corresponding to the value of the weighted function sum portion of the above equation [23]) is updated.

When this update is completed, the process proceeds to "BOX10", and the partial sum d of the function value f updated in "BOX9" is d.
The inner integral value T is updated according to T and the previously calculated inner integral value (corresponding to S _{k- 1} in the above-mentioned [Equation 23]). At this time, the inner integrated value T is saved in the specified data area area for use in the next update processing of the inner integrated value.

The process of updating the inner integral value is "judgment B".
Therefore, the process is executed until the setting of the interval dividing points of the inner integral specified by the calling source is confirmed. In this manner, the two-dimensional numerical integration calculation apparatus 1 executes the processing flow shown in FIG. 9 to set the number of inner division interval division points designated by the calling source and set the inner integration value to the inner integration value. It processes so that it may be calculated.

Next, the processing flow of FIGS. 7 and 8 will be described. By executing this processing flow, the two-dimensional numerical integration calculation device 1 doubles the interval division points of the outer integration when the two-dimensional numerical integration values have not converged. Then, the newly set section division points of the outer integration and the already set section division points of the outer integration are specified as the calculation targets of the inner integration, and the section division points of the outer integration are set as the points of the section division points of the inner integration. By designating the number of points and calling the subroutine shown in the processing flow of FIG. 9, the inner integral values for the interval division points of those outer integrals are acquired. Then, the two-dimensional numerical integration value that converges is calculated by calculating the outer integration value using the acquired inner integration value and determining whether or not it has converged. Here, when the subroutine shown in the processing flow of FIG. 9 is called, the inner integral value T acquired at the time of the previous call is notified so that this subroutine can calculate a new inner integral value in accordance with the update processing. Will be processed.

Specifically, the "BOX" in this processing flow is
1 ”to call the subroutine shown in the process flow of FIG. 9 using the central point of the outer section as a parameter to newly acquire the inner integral value T for this section dividing point, and also to obtain the outer integral section. This inner integral value T is updated by calling the subroutine shown in the processing flow of FIG. 9 in response to the increase in the number of division points.

Further, with "BOX2", the interval dividing points of the outer integration are doubled. Also, with "BOX3",
The inner integral value is updated by calling the subroutine shown in the process flow of FIG. 9 for the inner integral value calculated last time among the section dividing points of the doubled outer integral. In addition, with "BOX4", "BOX3"
The outer integral value is updated according to the inner integral value updated in.

In addition, by calling the subroutine shown in the processing flow of FIG. 9 for the "BOX5" among the interval division points of the doubled outer integral, for which the inner integral value has not yet been calculated, Calculate the inner integral value. Further, the outer integral value is updated by "BOX6" according to the inner integral value calculated in "BOX5". Then, the "judgment A" is used to judge the convergence of the outer integral value.

Next, according to the process explanatory diagram shown in FIG.
The processing of the processing flow shown in FIGS. 7 to 9 will be specifically described. As shown in FIG. 10, “i = 4” is set as the section dividing point of the outer integration at the first time, and correspondingly, “j =
The inner integral value is calculated by setting one inner integral section dividing point of "4", and the outer integral value is calculated using these inner integral values.

Next, at the second time, "i = 2, 6" is added in addition to "i = 4" as the interval division point of the outer integration, and correspondingly, each of these outer integrations is added. The inner integral value is calculated by setting three inner integral segment dividing points “j = 2, 4, 6” with respect to the segment dividing points, and the outer integral is calculated using these inner integral values. The value is calculated. At this time, since the line part consisting of the section dividing points of the black circles is the section dividing point of the inner integral defined by the new section dividing point of the outer integral, it is newly added without using the previous inner integral value. The inner integral value will be calculated. That is, it is necessary to newly obtain the function value for the black circle portion.

On the other hand, the line portion in which the section dividing points of the white circles and the section dividing points of the double circles are mixed is the section dividing point of the inner integral defined from the section dividing point of the outer integral set previously. Therefore, the inner integral value is calculated by using the inner integral value obtained based on the function values up to the previous time. That is, it is necessary to newly obtain the function value this time for the double circle portion, but it is not necessary to obtain the function value this time for the white circle portion.

Subsequently, at the third time, in addition to "i = 2,4,6", as a section dividing point of outer integration, "i = 1,3,3"
5, 7 ”is added, and correspondingly,“ j = 1, 2, 3, 4, ”for each of the interval dividing points of these outer integrals.
The inner integral values are calculated by setting the seven inner integral interval dividing points "5, 6, 7", and the outer integral values are calculated using these inner integral values.

As described above, according to the present invention, when the two-dimensional numerical integral value of the function is calculated, the inner integral is not separated from the outer integral and converged, but the score of the interval dividing points of the inner integral is calculated. And the two-dimensional numerical integration value are converged while controlling so that the number of section division points of the outer integration becomes equal.

[0108]

As described above, according to the present invention,
When calculating the two-dimensional numerical integration value of a function, the variable y
Is integrated first and then the variable x is integrated, or the variable x is first integrated and then the variable y is integrated. Therefore, it becomes possible to calculate a two-dimensional numerical integration value that can obtain the same value regardless of the order of integration.

Further, in the present invention, when the two-dimensional numerical integral value of the function is obtained, since a single convergence judgment configuration is adopted, two convergence judgment values must be set conventionally. The annoyance of technology can be avoided.

[Brief description of drawings]

FIG. 1 is a principle configuration diagram of the present invention.

FIG. 2 is an example of a processing flow of the present invention.

FIG. 3 is an example of a processing flow of the present invention.

FIG. 4 is an example of a processing flow of the present invention.

FIG. 5 is an example of the processing flow of the present invention.

FIG. 6 is an explanatory diagram of processing of the present invention.

FIG. 7 is an example of the processing flow of the present invention.

FIG. 8 is an example of the processing flow of the present invention.

FIG. 9 is an example of the processing flow of the present invention.

FIG. 10 is an explanatory diagram of processing of the present invention.

FIG. 11 is an explanatory diagram of a two-dimensional numerical integration process.

FIG. 12 is a processing flow of a conventional technique based on Simpson's rule.

FIG. 13 is a processing flow of a conventional technique based on the Simpson rule.

FIG. 14 is an explanatory diagram of a conventional technique.

[Explanation of symbols]

 DESCRIPTION OF SYMBOLS 1 Two-dimensional numerical integral calculation device 10 Function information management means 11 First calculation means 12 Section dividing point setting means 13 Storage means 14 Second calculating means 15 Judging means 16 Starting means 17 Section dividing point setting means

Claims (7)

[Claims] 1. A two-dimensional numerical integration calculating device having a configuration for calculating a two-dimensional numerical integral value of a given function, when a coordinate point of an inner integration target which is a section dividing point of outer integration is given. Set the section dividing points of the inner section that have a value equal to the number of coordinate points, and follow the specified integration method,
A first calculating means (11) for calculating an inner integrated value of two-dimensional numerical integration using the function value of the section dividing point, and when the first calculating means (11) calculates the inner integrated value, A second calculation means (14) for calculating an outer integration value which is a two-dimensional numerical integration value using the inner integration value according to a prescribed integration method, and an outer integration value calculated by the second calculation means (14). When the judgment means (15) for judging the convergence of the value and the above judgment means (15) judge the non-convergence, the interval dividing point of the outer integration having a larger value than the previous time is set to determine the inner integration target Set a new coordinate point
And a starting means (16) for activating the calculating means (11). 2. The two-dimensional numerical integration calculation device according to claim 1, wherein the starting means (16) sets a new coordinate point to be an inner integration target by newly adding an interval division point of the outer integration. A two-dimensional numerical integration calculation device characterized by performing processing so as to activate the first calculation means (11). 3. The two-dimensional numerical integral calculation device according to claim 2, wherein the inner integral value calculated by the first calculating means (11) and the intermediate calculated value used for the calculation processing of the inner integral value are calculated. The storing means (13) for storing while updating is provided, and the starting means (16) is provided with the corresponding inner side stored in the storing means (13) for the coordinate point of the inner integration target for which the inner integral value has already been calculated. The integrated value and the intermediate calculated value are used as notification information to start the first calculating means (11), and the first calculating means (11) notifies the inner integrated value and the intermediate calculated value from the starting means (16). A two-dimensional numerical integral calculating device characterized by performing processing to calculate a new inner integrated value and intermediate calculated value with higher accuracy using the inner integrated value and intermediate calculated value. 4. The two-dimensional numerical integration calculation device according to claim 3, wherein the first and second calculation means (11, 14) are Si integration methods.
A two-dimensional numerical integral calculation device characterized by processing using the mpson rule. 5. The two-dimensional numerical integration calculation device according to claim 2, further comprising a storage means (13) for storing the inner integration value calculated by the first calculation means (11) while updating the start-up means (16). ), For the coordinate point of the inner integration target for which the inner integration value has already been calculated, the corresponding inner integration value stored in the storage means (13) is used as notification information for the first calculation means (1
In addition to activating 1), when the first calculating means (11) notifies the inner integral value from the activating means (16), it uses the inner integral value to generate a new inner integral value with higher accuracy. A two-dimensional numerical integration calculation device characterized by performing processing to calculate. 6. The two-dimensional numerical integration calculation device according to claim 5, wherein the first and second calculation means (11, 14) perform processing so as to use a trapezoidal rule as an integration method. Dimensional numerical integration calculator. 7. The two-dimensional numerical integration calculating device according to claim 5, wherein the first and second calculating means (11, 14) are I as an integration method.
A two-dimensional numerical integration calculation device characterized by performing processing using the MT rule.