JPH10307810A - Method for displaying solution procedure of simultaneous equations and computer-readable recording medium recording program for displaying the solution procedure - Google Patents

Abstract

PROBLEM TO BE SOLVED: To support operation for assembling a calculation procedure while grasping relation between respective equations constituting simultaneous equations and variables included in the equations and to display a practical calculation procedure while automatically checking relation between the equations and variables. SOLUTION: An area 1 corresponds to the names of equations, an area 2 corresponds to the names of variables, an area 3 is an area for a flag specifying the participation of the equations and the variables, and an area 4 is the area of a flag specifying whether a variable has been known or not. An area 5 is the area of a three-state flag for demanding definition and areas 6, 7 are prepared for moving rows and columns. The setting of existence of particiation indicating whether respective equations include their individual variables or not and definition indicating the presimulation of a procedure for executing the simultaneous calculation of the set equations ate displayed in the area 3 by two symbols.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for displaying a solution procedure for a system of simultaneous equations and a computer-readable recording medium on which a program for displaying a solution procedure for a system of simultaneous equations is recorded. It can be widely used in the field of dealing with phenomena and in the field of education.

[0002]

2. Description of the Related Art Simultaneous equations have been mainly used for a long time by calculation by a logicist in the field of science and technology.

However, recently, with the progress of computer science, it has become possible to solve the problem by a mathematical method or a numerical method. Although software products created for such purposes are commercially available, many products identify variables by name, so they are linked to each other through expressions, and through variables. It is difficult to express the simultaneous structure such as the connection between the formulas with good visibility.

Another typical solution by a computer is a simultaneous solution in which conditions are given by a large matrix. This is especially the most studied and exploited in the case of linear equations.

In addition to the linear equation, Newton's equation
Non-linear calculations are also performed by Rafson method, least-squares method, etc., but in these non-linear calculations, the convergence becomes worse as the number of simultaneous variables and equations increases, and the calculation time increases. There is a serious problem.

[0006]

By the way, in the simultaneous equations, most of the individual equations have a relation only with a very small part of the variables used in the whole system. Very often it has a sparse shape. Therefore, even when aiming for a numerical solution, it is efficient to first grasp the simultaneous structure of the solutions, design a reasonable calculation procedure, and then start the numerical calculation. Accurate accuracy can be estimated.

However, at present, software which is easy to handle for logically grasping and calculating the simultaneous structure in this manner is not provided at present.

The present invention has been devised to solve such a problem, and has as its object the above tasks conventionally performed by the brain, that is, the equations constituting the simultaneous equations and the equations contained therein. In addition to assisting in assembling the calculation procedure after understanding the relationship with the variables to be
Provided are a method for displaying a solution procedure of a system of simultaneous equations and a computer-readable recording medium on which a program for displaying a solution procedure of a system of simultaneous equations is recorded, which can show the actual calculation procedure while automatically confirming the relation between equations and variables. Is to do.

[0009]

In order to solve the above-mentioned problems, a method for displaying a solution procedure of a system of simultaneous equations according to claim 1 of the present invention comprises a display means capable of redrawing and input of a position of two or more dimensions. A method for displaying on a screen of the display means a logical procedure for deriving the value of each variable of the simultaneous equation system through interactive input / output, wherein each equation includes an individual variable It is composed of two stages, namely, setting of the presence or absence of involvement indicating whether or not, and determination as a pre-simulation of a procedure in which simultaneous calculation is performed on the equation for which this setting has been made. The equations are arranged in rows, and a matrix in which each variable is associated with a column is displayed on the screen of the display means. The symbols of the empty elements are displayed at the intersections on the non-existent matrix, and when a certain set A made up of the equations is arbitrarily selected by the operation of the input means, one or more of the selected sets A are selected. It is determined whether or not the set B formed by all of the undetermined variables involved has the same number of elements. If the result of this determination is that the set A and the set B have the same number of elements, the undetermined variable Is symbolized as a deterministic variable, and the equations included in the set A are symbolized as deterministic, and the equations other than the set A include the deterministic variable.
Change the symbol of the intersection on the matrix corresponding to the determined variable to a symbol indicating involvement and display it. Then, for equations other than the set A, select the set, perform the determination, and display the determination and involvement. By recursively repeating the process of performing the evaluation by (1), all equations are displayed by symbols of determination and involvement.

A computer-readable recording medium storing a program for displaying a solution procedure of a system of simultaneous equations according to a second aspect of the present invention sets the presence or absence of involvement indicating whether or not each equation includes an individual variable. And a determination, which is a pre-simulation of the procedure in which the simultaneous calculation is performed for the equation for which this setting has been made, and includes a line for each equation of the system of equations to be processed, and a column for each variable. Is displayed on the screen of the display means, a symbol of a filling element is provided at an intersection on the matrix where a variable exists in each equation, and an empty element is provided at an intersection on the matrix where no variable exists in each equation. When a set A for which an equation is to be created is arbitrarily selected by the procedure for displaying the symbols and the operation of the input means, one of the selected sets A is selected. A procedure for determining whether or not the set B created by all of the undetermined variables involved with the above has the same number of elements. As a result of this determination, if the set A and the set B have the same number of elements, Represents the undecided variable as a deterministic variable, and the equations included in the set A as the determinant. For equations other than the set A that include the deterministic variable, A procedure of changing the symbol of the intersection on the corresponding matrix to a symbol indicating involvement and displaying the same, and then, for the remaining equations other than the set A, selecting the set, performing the determination, and displaying the determination and the involvement A procedure of recursively repeating the evaluation process by and a procedure of displaying all equations by symbols of confirmation and involvement by recursively repeating,
In a computer.

[0011]

Embodiments of the present invention will be described below with reference to the drawings. FIG. 1 shows typical functional divisions of a screen displayed by a method of displaying a solution procedure of a system of simultaneous equations according to the present invention.

That is, the area (A) and the area (A) are for inputting and displaying characters, and (A) corresponds to the name of an expression and (A) corresponds to the name of a variable. . By clicking on these with a mouse (not shown) (more precisely, a pointer moved by the mouse), character input can be accepted, and the name can be input using a keyboard (not shown).

Each of the areas (c), (d), and (e) is an area in which a flag indicating the status of the system is displayed and a change in the flag is accepted through the mouse. The area (c) is a flag area for designating the relationship between the equation and the variable. By clicking on the flag, the presence or absence of the relationship can be changed. However, this does not apply in the case of prohibition (for example, when the equation is already determined).

The region (d) is a flag for designating whether or not the variable is known, and is alternately switched between the two states by clicking the mouse.
The area (e) is a three-state flag for requesting confirmation, and is operated with a mouse. By operating the flag one by one, the first flag (0) is set for each of the requested flags. To the second state (1). Then, when the combination of claimed equations becomes sufficient to make a determination, the state changes to the third state (2).

The areas (f) and (g) are areas for moving rows and columns. As for the area (f) relating to the row, when the mouse is moved over this and the button is pressed, the button is activated, and the color of this portion of the corresponding row changes. If you hold down the button and move it up or down, it will be switched to the destination line one after another, and the first activated line will move so as to follow the mouse (pointer). This movement is (a),
It occurs simultaneously for each of the regions (c) and (e).

The area (g) relates to the movement of the row.
The function is the same as that of the region. This movement is (a),
It occurs simultaneously for each of the areas (c) and (d). Here, the names of the variables written in the area (a) are as shown in FIG.
As shown in (a) and (b), the display is stepwise. When changing the number of steps, the area shown in FIG. That is, when the mark connected to the horizontal line is caught with the mouse and moved up and down, the number of steps changes. FIG. 2A shows the case where the number of steps is five, and FIG. 2B shows the case where the number of steps is four. When the number of steps is changed from five to four, the data is written in the lowermost part on the left side. The variable name (pressure 1) has been moved to the uppermost row on the right. As the number of columns increases, longer names can be used, but it is difficult to see if the name goes too high, so select an appropriate number of columns according to the purpose.

[Technical background of the method of displaying the solution procedure of a system of simultaneous equations according to the present invention] In general, a simultaneous equation including n variables usually determines the entire variable value only when n equations are simultaneously used. can do. However, it is not always possible to know one variable until all n equations are applied. In a real problem, it is rather possible to know the same number of variables as the number of n parts by a simultaneous system of n parts. Often you can.

Also, in the case where it is determined for the first time by applying all the equations, by assuming only a few of them, the remaining variable values are derived, and only the previously assumed variables are independent. In some cases, it can be reduced to a system of equations based on variables.

If this can be easily foreseen, the convergence calculation can be performed based on fewer variables, which is convenient. The present invention is suitably used for examining situations between such equations and variables and designing calculations.

In general, the simultaneous equations are generally determined by the same number of equations as the number of unknown variables. Of course, the solution may not actually exist or may not be uniquely determined.However, in a system of analytic functions that defines continuous events in the natural world, it is a special case that the solution is not uniquely determined. is there.

In the present invention, special cases depending on the individual assembly of functions and operations are ignored, and the logical order in which the values are determined is first followed by focusing only on the number of simultaneous variables and expressions. The policy is.

The operation of the software of the present invention is based on whether each equation includes an individual variable or not, that is, the setting of "involvement", and the advance of the procedure in which simultaneous calculations are executed for the equation for which this setting has been made. It consists of two stages of simulation, that is, "determination".

That is, each equation of the system of equations to be processed is applied to a row and each variable is applied to a column, and the presence or absence of “involvement” between them is represented by the distinction of symbols at intersections. In other words, the case where there is no "participation" is referred to as "empty element", and the case where there is no "participation" is referred to as "filling element". And
Within them, a distinction is made according to the rules described below. As a result, in the present embodiment, three symbols (•, |,）) and “filling element”
Are represented by four symbols (o, ◎, Φ, *), respectively.

[Regarding the kernel (irreducible system) and its hierarchy] A system of n equations including n unknown variables determines those unknown variables by simultaneous system, but the partial variables and the partial expressions If the system is not determined, the n systems of equations are said to be "irreducible". In the software of the present invention, it is a principle that an equation system is constructed only by irreducible determination. The unknown variable is determined by an irreducible system and becomes a "known" variable.

Now, the irreducible n equations and n variables determined at the same time, and the entirety of the n × n filling elements and empty elements at the intersections of these equations are referred to as “nuclei”. Filled elements belonging to the core are marked with “◎” and empty elements are marked with “○”
It shall be represented by Equations that once belong to the nucleus are not reused for later determination, and the variables are also known variables, so they are not reused as unknown variables. Therefore, the same equation, variable, and element cannot belong to two or more different nuclei. The filling element in this case is represented by (Φ).

With such a filling element, if any variable belonging to the previously determined nucleus D has a relationship with any equation belonging to the subsequently determined nucleus E, then the nucleus D is said to be the "senior" of the nucleus E. Name it. In this case, the variables previously defined by the action of the nucleus D act as known constants in the nucleus E (ie, they are not unknown variables) and give meaning to the simultaneous equations of the nucleus E. Because it is.
Conversely, kernel E is a "junior" of kernel D.

Therefore, if the kernel D is a senior of the kernel E and the kernel E is a senior of the kernel F, the kernel D is an indirect senior of the kernel F. In this way, seniors and juniors have a direct and indirect connection of the filling element at the intersection by Φ, but conversely those that have such a connection indirectly
There is not always a relationship between seniors and juniors.

For example, if nucleus D is a direct ancestor of nucleus E and nucleus F and nucleus E and nucleus F have no mutual involvement,
There is an indirect connection between the nucleus E and the nucleus F, but they are not senior and junior relationships. Also, if both nuclei D and nucleus E are direct seniors of nucleus F and nuclei D and nucleus E have no mutual involvement, again, which is the senior between nuclei D and nucleus E But I'm not a junior.

The involvement between nuclei can be switched, such as adding or removing. However, cyclical designation of two persons, direct or indirect, senior and junior, such as establishing a reverse senior and junior relation without excluding this relation, is prohibited. If this is allowed, the scheme for solving the equations sequentially breaks down. If such forms of engagement do exist, then a large unit system that includes these engagements must be considered.

[Regarding element operation and function using mouse] In the operation of the software, when the position of the matrix element of the undetermined row [(f) in FIG. 1] is clicked,
Basically, an empty element “•” (or “|”) changes to a filling element “o”, “Φ”, “*”, and returns to an empty element in the case of a filling element.

The elements at the intersections where the rows and columns respectively belong to the two nuclei determined individually can be switched between the empty element and the filling element, but in the case of the filling element,
An automatic check is performed to prohibit the above-mentioned cyclic designation, and if this is violated, a warning message is issued and the contents are not changed.

[Operation and Function of Constant Flag Using Mouse] In software operation, when clicking on the constant flag ((D) in FIG. 1), basically, the non-constant "." The constant "C" is a non-constant "."
become. The column (variable) in which the constant “C” is set is treated as a fixed variable.

If a constant flag is clicked on a column in which an already determined element exists, the column is not converted to a constant. In order to make this a constant, it is necessary to first release the determination. Conversely, when the constant is released, if there is a filler element in the column, the content changes from "Φ" or "*" to "o". However, if it is “Φ” (when it is involved in the determined row), the determination of the nucleus including that row is canceled. This cancellation is performed recursively for direct and indirect juniors.

[Regarding the Function of Operating the Confirmation Flag Using the Mouse] When the button for confirmation ((E) in FIG. 1) is clicked, the color of the button changes and the flag value changes from 0 to 1. When the set of all or parts of the rows having the flag value of 1 reaches a determinable state (when only the same number of columns are involved as unknown variables), the smallest irreducible of them is shown. The final set is determined.
The flag of the determined line changes to 2. The re-evaluation for this is performed not only when the flag is newly changed, but also when the matrix element “o” of the undefined column (variable) is removed (ie,
This is also performed when the value becomes “•” and when the constant flag is set (that is, when the value becomes “C”).

If the line set by the flag operation has already been determined (the flag value is 2), this determination is released. This cancellation is performed recursively for direct and indirect juniors.

The line clicked is in the first state,
If there is no "o", no response occurs.
If the user clicks on a button whose flag value is 1, the confirmation flag is simply returned to 0.

[Movement of Rows by Mouse] If the user moves the mouse cursor while clicking the position (f) in FIG. 1, the entire column, that is, the name, constant flags, and matrix elements move according to the mouse. The (next) row at the destination is replaced by the row where the mouse pointer was first. This replacement is
At the same time as the display, internal data is also displayed.

[Changing the Number of Variable Names Using the Mouse] By clicking and moving the position (G) in FIG. 1, the number of variable names can be changed. For example, if the number of stages is reduced by one from the state of FIG.
become that way.

Next, a specific example of a program configuration for realizing the above functions will be described. The example of the program configuration has, for example, a functional block configuration as shown in FIG. 3, and each block functions according to the above-described individual processing items. That is, "input of expression name" can be input by clicking (a) of FIG. 1 with a mouse, and "input of variable name" can be input by clicking (a) of FIG. 1 with a mouse. "Change flag" can be changed by clicking (c) in Figure 1 with the mouse.
"Constant flag setting" can be set by clicking (e) in FIG. 1 with a mouse, and "change of final flag" can be changed by clicking (e) in FIG. 1 with a mouse. "" Can be moved by moving the mouse up and down while clicking on (f) in Fig. 1. "Move column" can be moved by moving left and right while clicking (g) on the mouse in Fig. 1, The "change of variable name stage number" can be changed by clicking (h) in FIG. 1 with a mouse and moving it up and down.

For data, a variable i is applied to each row (expression) and a variable j is applied to each column (variable). As the one-dimensional array data for the row i, the name of the expression and the confirmation flag (an integer from 0 to 2) are placed. A change flag is also placed as one-dimensional array data for row j, which is primary data that should be held only during each response. As the one-dimensional array data corresponding to the column j, a variable name and a constant flag (the expressions of “•” and “C” are distinguished by 0 and 1) are placed.

A filling flag and a kernel flag are set as two-dimensional array data corresponding to the matrix (i, j). The filling flag is an integer of 0 or 1, where 0 represents an unfilled element and 1 represents a filled element. The nuclear flag indicates a non-nuclear element by 0 and a nuclear element by 1. As a combination of (filling flag, nuclear flag), (0, 0) is replaced with “•” or “|”, (1, 0) is replaced with “o”, “Φ”, or “*”, and (0, 1). Changed to "○",
(1, 1) respectively correspond to “◎”.

The following three subroutines are the main processing subroutines.
Put one thing. These are further called from the routines described below. 1. Construction routine Takes no arguments. From the whole or a part of the set of rows whose determination flag is 1, a searchable one is searched for and determined. That is, the intersection between the n rows and the same n columns is "nucleus" (nucleus flag is 1), and the decision flags of these rows are 2. Therefore, the fact that a row has a nucleus (an element whose nucleus flag is 1) and the fact that a row has a fixed flag of 2 always occur simultaneously. If any changes are made in response to the call, a recall is made.

2. Search Routine Takes an argument i corresponding to a row and uses a change stack that stores the set of i for output. Generally, the contents of the change flag are cleared as a procedure prior to calling the search routine. If this line i is determined for the call that specified i, the search routine sets the i-th element of the change flag to 1. Rows i 'having a non-nucleus filler element at the intersection with any one of all js forming a core with i are searched one after another, and the search routine itself is recursively called using these i's as arguments. In the call at this time, the change stack is not cleared in advance.

As a result, a representative row i of each kernel which becomes a direct or indirect junior to the first given row i is obtained as the contents of the change stack. 3. Release Routine In calling this routine, the contents of the change flag described above are passed. In this routine, the determination is canceled for the nucleus related to each line having the contents of the change stack (that is, the determination flag of the same line as all the kernel flags of the specified lines is set to 0).

In the specification from the mouse or the like, the first called routine is as follows. [Decoding of instructions and arranging processing] In the case of procedural programming, the designation of processing by operating a mouse or the like is to read the position of the mouse and the state of the button with an input sentence placed in an infinite loop, and A method for calling individual processing. Also, in an object-oriented language system, a process is started by associating a mouse operation or the like with a so-called event and associating the event with a desired individual function.

1) Input of the name of the formula: The formula is activated by clicking at a position ((A) in FIG. 1) corresponding to each name. 2) Input of the name of a variable: It is started by clicking at the position corresponding to each name ((a) in FIG. 1).

3) Switching between involvement and non-participation: Activated by clicking on a confirmation flag [(c) in FIG. 1]. 4) Setting and canceling of a constant: The process is started by clicking on a line dedicated to a constant [(D) in FIG.

5) Switching of the confirmation flag: This is activated by clicking on the confirmation flag [(e) in FIG. 1]. 6) Movement of line (expression): Activated by clicking on (f) in FIG. 1 and responds in response to movement.

7) Movement of column (variable): Activated by clicking on (g) in FIG. 1 and responds in response to the movement. 8) Acceptance of other commands: Commands such as file operations are accepted by a pull-down menu or the like of the window system.

When the processing is completed at each branch destination, the display is changed in response to the change of the internal data, and then the display returns to the initial state again to receive an input. The following are directly called from the above-mentioned routines related to the decoding of commands and the arrangement of processing, and further utilize the above-described routines such as construction, search, and cancellation as necessary to perform processing.

[Input of Name] The name at the position designated by the mouse operation is input through the keyboard.

[Switching of Involvement or Not Involvement] (C) in FIG.
Is clicked, the two-dimensional position is converted into a set of numerical values corresponding to rows and columns, and passed to a processing routine. The data of the presence or absence of the region (c) is stored in the program in an array form separately from the display on the screen, and the processing flow is divided accordingly.

First, if the specified element is not a filling element, a change in the direction in which this is the filling element may occur.
Rows that have not been determined (the determination flag is not 3) are unconditionally converted into filling elements. Also, for a row that has been confirmed, the specified column is confirmed at another point in time, and after confirming that the determination of the column is not a junior of the row determination through the search routine described above, the filling is performed. It is elementized. In addition,
If the specified element is the core, if the row is fixed but the column is not fixed, or if both the row and column are fixed but the cycle occurs, Does not change.

If the specified element is a filling element, if it is a nucleus, nothing changes. If it is not a nucleus, make it an unfilled element. If the confirmation flag is 1 (that is, if the specified element is not always the core),
Thereafter, a re-evaluation is performed to determine if the set containing the row is not feasible (this re-evaluation is based on the construction routine described above).

[Setting and Canceling of Constants] If the designated one is not a constant from the beginning, it is converted into a constant only when the column is not determined, and then the above-mentioned construction routine is called.

If the specified one is a constant, it is unconditionally changed to a non-constant. After clearing the change stack, all rows having a filling element on the specified (constant) column are searched. The routine is called (the change stack is not cleared halfway), and then the release routine is called.

[Switching of Confirmation Flag] The confirmation flag is
It takes three states, 0, 1, and 2. If it was originally 0, it is changed to 1 and the construction routine is called.

If the finalization flag is originally 1, it is simply returned to 0. If the decision flag is 2, this line is designated, the change stack is cleared, the search routine is called, and then the release routine is called.

[Move Line (Formula)] When the mouse is pressed on the area (f) in FIG. 1, a routine for moving a line is entered. If a movement across a line boundary occurs while the button is pressed, the line at the destination (immediately above or immediately below) is replaced with the original line. At this time, not only the screen
The contents of the name of the dimensional expression, the confirmation flag, the two-dimensional filling flag, and the kernel flag are also replaced at the same time. Such a response continues until the press is released.

[Movement of Column (Variable)] As in the case of row movement, column movement occurs. However, the names of the rows are stepped as shown in FIG. 2, so that the movement is performed according to this. In addition to the screen, the contents of the one-dimensional variable name, the constant flag, the two-dimensional filling flag, and the kernel flag are simultaneously replaced. Such a response continues until the press is released.

[Reception of other commands] The minimum necessary data for inputting / outputting a file is one-dimensional array data such as an expression name, a variable name, a constant flag, and a decision flag, and a two-dimensional array of a filling flag and a nuclear flag. And data.

Next, a method of displaying a solution procedure of a system of simultaneous equations realized by each of the above functions will be specifically described by exemplifying a plurality of types of actual simultaneous equations. [Example 1] Example 1 is a quaternary system of equations expressed by the following equations (1) to (4), describing the relationship between a constant a and variables w, x, y, and z.

[0063]

X = a ² (1) xy + z = 1 (2) y−2z = 1 (3) α = z + 3 (4) From the equations (1) to (4), finally, α = Considering the procedure for deriving a ² +1 (5), substituting x by equation (1) into equation (2), and simultaneously with equation (3), z = a ² -2 (6) A method of deriving equation (5) from equation (4) can be considered.

In addition, first, the equations (2) and (3) are made simultaneous, z = x−2 (7), and the equation (6) is derived from the equations (1) and (7). After substituting formula (4) into formula (7), formula (1) is used, or formula (4) is substituted into formulas (2) and (3), and z is first eliminated. Various routes such as a method are conceivable. In this way, how many proof paths are related to each other is not very clear as long as it is proved using an expression as in the past,
As explained below, the method according to the invention makes it possible to see clearly.

That is, the relationship between the equation (row: each of the horizontal rows) and the variable (the column: each of the vertical rows) is expressed by “o” when a variable exists in each equation, and “•” when it is not. To express,

[0066]

[Table 1]

Is expressed as follows. The right end of each line is the value of what is named a fixed flag, which takes one of integer values of 0, 1, and 2. Since a is a constant among the above variables, this is first expressed. That is, the variable names (a, x,
The line immediately below (y, z, α) is a special line for a flag indicating a constant, and this element takes two values, “•” and “C”. “C” is an expression indicating a constant, and “•” is an expression indicating that it is not. Therefore, a is a constant,

[0068]

[Table 2]

Is shown. At the same time that C is raised, "o" in the same row has changed to "Φ". This graphically represents that a vertical line | is drawn on “o” because the variable has become a constant and need not be counted as an unknown variable.

Since the equation (1) consists of this and only the variable x, it is determined next. This is done by placing the mouse pointer on the rightmost confirmation flag and clicking on it. At this time, the confirmation flag changes from 0 to 1 once, but at this time, the number of rows where the confirmation flag is 1 is 1, the number of unknown variables involved in the same row is also 1 and both are the same 1. Therefore, the decision takes place. as a result,

[0071]

[Table 3]

The decision flag is further changed from 1 to 2. “◎” indicates an involved element that has been determined. Another "o" in the same row
One has been changed to "Φ", which indicates involvement.

Next, when the mouse pointer is placed on the final flag at the right end of each line in the expressions (2) and (3) and clicked, the final flag of these lines changes from 0 to 1 once. At this point, the number of rows having the confirmed flag of 1 is 2,
The determination occurs because the number of unknown variables (y, z) involved in these rows is also 2, and both are the same. As a result, when the simultaneous equations (2) and (3) are determined,

[0074]

[Table 4]

Thus, the decision flag of each line changes from 1 to 2. In addition, “o” in each column of y and z, which were unknown variables, has been changed to “◎” which indicates an element involved in the determination. Another thing in the same z column that was “o” has been changed to “Φ” indicating involvement.

Next, when the mouse pointer is placed over the finalization flag at the right end of the line in equation (4) and clicked, the finalization flag of this line changes from 0 to 1 at this point. Since the number of rows that are 1 is 1 and the number of unknown variables (α) involved in these rows is also 1 and both are the same, the determination occurs. as a result,

[0077]

[Table 5]

The following result is obtained. This proves that the value of α can be obtained anyway. Of course, this is aside from the existence of solutions that depend on individual equations, and the problem of uniqueness. In addition, it is understood that the solution is likely to be obtained, but not the solution itself. However, the specific procedure for obtaining the solution depends on Table 5 finally obtained by the above processing. be able to.

[Example 2] Example 2 shows a case in which a variable that has not been determined in Example 1 but has already been determined appears in a variable that has not yet been determined and an equation that does not have any of the variables determined therein. The following (11) to (1)
4) It is a quaternary simultaneous equation system by the equation. The equation in this case is represented by “*”.

[0080]

## EQU2 ## a = 1 (11) x = a (12) x = 2a-3 (13) y = x + 1 (14) This simultaneous equation is an inconsistent simultaneous equation and has no solution. It is expressed as

[0081]

[Table 6]

Since a is a constant among the above variables, first, when formula (11) is determined,

[0083]

[Table 7]

Is obtained. That is, the fixed flag of the line changes from 1 to 2. In addition, “o” in the column of “a”, which has been treated as an unknown variable, has been changed to “表 す” representing the determined one. In addition, the other two items in the same column a, which have been “o”, have been changed to “Φ” indicating their involvement.

Further, focusing on the fact that a is a constant,
The following expression can be used instead of [Table 7].

[0086]

[Table 8]

That is, the flag of the column a in the row immediately below the variable name (a, x, y) is changed to “C”. Then, at the same time that C is set, “o” in the same row is changed to “Φ”. The process proceeds further based on [Table 8]. That is, (1
2) Since the equation consists of this and only the variable x, this is determined next. This is done by placing the mouse pointer on the rightmost confirmation flag and clicking on it. At this time, the confirmation flag changes from 0 to 1 once,
At this point, the number of rows having a determination flag of 1 is one, and the number of unknown variables involved in the same row is also one. in this case,
No unknown variables remain in the equation (13). As described above, when the unknown variable has become zero, although it has not been determined, the filling element of the row is represented by “*”. as a result,

[0088]

[Table 9]

## EQU11 ## Here, it is suggested that equation (13) is an extra equation from the point of determination of the solution, and that if it is fixed, there may be no solution. Therefore,
If equation (14) is determined without worrying about this, the result is

[0090]

[Table 10]

Thus, “*” does not disappear.
The presence of "*" anywhere in the table indicates that the part is over-defined and can be inconsistent.
Here, once again summarizing the above expression, "o",
“◎”, “Φ”, “*” are filler elements indicating that the variable exists in the equation, that is, there is a relationship between the equation and the variable. Empty element.

[Example 3] In Example 3, the relationship between variables x, y, z, and w is described, which is expressed by the following equations (21) to (24).
It is a system of original simultaneous equations.

[0093]

Xz = 2 (21) x + y + z = 6 (22) xy = 1 (23) w = z + 1 (24)

[0094]

[Table 11]

Is represented by Here, equation (21), (2
When equations (2) and (23) are combined, ie, equation (21)
When the mouse pointer is placed on the final flag at the right end of each line of the formulas (22) and (23) and the mouse pointer is clicked sequentially, the final flag of these lines changes from 0 to 1 at this point. Since the number of rows where the determination flag is 1 is 3 and the number of unknown variables (x, y, z) involved in these rows is also 3 and both are the same, determination is made. As a result, equations (21), (22), (2)
3) When the simultaneous equations are determined,

[0096]

[Table 12]

Thus, the decision flag of each line changes from 1 to 2. In addition, “o” in each column of x, y, and z, which were unknown variables, has been changed to “表 す” that indicates the element determined as an involved element. Another thing in the same z column that was “o” has been changed to “Φ” indicating involvement.

Here, “○” means that the corresponding equation is a member of a system of simultaneous equations defining variables, but that the corresponding variable is not included in the corresponding equation itself. Although the use of “•” does not cause logical confusion in the meaning of “○”, it has been empirically found that the use of “○” makes it easier to see. You are. “O” is also an empty element.

Example 4 Example 4 is a ternary system of equations expressed by the following equations (31) to (33), describing the relationship between variables x, y, and z.

[0100]

X + y = 1 (31) y = 2 (32) y + z = 3 (33)

[0101]

[Table 13]

Are represented as follows. Here, in order from the top (31)
When the expression, the expression (32), and the confirmation flag are operated, the expression (32) alone alone sufficiently defines the variable y.
This is determined first because it is an irreducible equation. As a result, the confirmation flag of the line changes from 1 to 2. Also, “o” in the column of y, which was treated as an unknown variable in the row of the equation (32), has been changed to “表 す” representing the determined one. Also, the other two in the same y column, which were “o”, have been changed to “Φ”, which indicates their involvement. Therefore, equation (31) is determined next, and the result is

[0103]

[Table 14]

Is obtained. Subsequently, when the expression (33) is specified,
Since the equation (33) consists only of the determined variables y and z indicating the involvement, this is determined next. That is, the user places the mouse pointer on the confirmation flag at the right end and clicks. At this time, the confirmation flag changes from 0 to 1 once, but at this time, the number of rows for which the confirmation flag is 1 is 1,
The determination occurs because the number of unknown variables involved in the same row is also one and both are the same. as a result,

[0105]

[Table 15]

The following is obtained. Rearranging the order for equations and variables,

[0107]

[Table 16]

Is obtained. Expressing in this way, first (32)
Y is determined by the equation, and x is determined by the equation (31).
Is determined, and finally, the process in which z is determined by equation (33) is clearly understood.

Here, the fact that two irreducible simultaneous equations cannot be made at the same time will be proved with reference to FIGS. 4 (a) and 4 (b). If the equations are specified one by one according to a certain order, only one irreducible system of simultaneous equations appears at first, and two simultaneous equations cannot be created at the same time. If two irreducible simultaneous equations systems G and H are formed at the same time, both G and H should be square matrices. So the line (equation)
Is compared with the number q of common parts of columns (variables), and if p <q, the sum (combined) of the equations G and H is the last one specified Contradiction [Fig. 4]
(A)], if not p ≧ q, then the product (intersection) of the system of equations G, H is irreducible if G and H are equal or smaller at the latest by the participation of the last equation. Although a system of equations should be created, from the initial assumption that G and H are irreducible, this must be G and H themselves, that is, G and H must be the same [Fig. (See (b)). This proved that two irreducible simultaneous equations systems cannot be formed at the same time.

[Example 5] In Example 5, the illustration of a concrete system of simultaneous equations is omitted, and the processing is performed in the same manner as in Examples 1-4 above.
Only the finally obtained result is illustrated in Table 17.

[0111]

[Table 17]

Table 17 may be left as it is, but y and z are determined by simultaneous equations by the equations (41) and (42).
The z determined here and x for which the constant has been specified are:
In order to emphasize that it is also used in the simultaneous equations (43) and (44), a symbol "|"

[0113]

[Table 18]

Then, it becomes easier to see. "|" Itself is an empty element. In summary, “•”, “O”, and “|” are empty elements. The display rule of “|” indicates that all of the filling elements or “O”, from the one located at the bottom of each column to the one located at the top of each column (“C” for a constant) Rewriting empty elements (except for “「 ”) from“ • ”to“ | ”makes them easier to see.

[Example 6] Next, a calculation regarding the network shown in FIG. 5 will be described. In addition to the power supply voltage V, each of the resistors R1 to R5
Is given by Ohm's law regarding the resistances R1 to R5 and the condition that the sum of the inflow current amounts at the points P6, P7 and P9 is zero, the following simultaneous relationship is established.

[0116]

[Table 19]

The main "nucleus" part of this equation is a seven-element simultaneous linear equation. Although it is possible to calculate as it is, here, assuming that V1 is a constant without knowing the value,
Table 20 can be obtained by appropriately changing the order of the remaining variables and expressions. According to this table, the formulas can be applied in order from the formula located at the top. Here, over the condition of P7 (inflow to point P7 = 0), a one-dimensional linear equation having only V1 as an indeterminate factor can be obtained. This can be easily solved, and based on the obtained V1. , All variables can be obtained again according to the procedure shown in Table 20.

[0118]

[Table 20]

As described above, by using the method for displaying the solution procedure of the system of simultaneous equations according to the present invention, constants, variables,
And the equations including them are listed in a prospective manner, so that the procedure for finding the solution is easy and has few errors.

The image processing for displaying the solution procedure for the simultaneous equation system is realized by a solution procedure display program for the simultaneous equation system, and this program is provided by being recorded on a recording medium.

[0121]

According to the first aspect of the present invention, there is provided a method for displaying a solution procedure for a system of simultaneous equations, wherein the setting of presence / absence indicating whether or not each equation includes an individual variable, and the equation for which this setting has been made are provided. The display means includes a matrix in which each equation of the system of simultaneous equations to be processed is arranged in a row and each variable is associated with a column. And a symbol of a filling element is displayed at an intersection on a matrix where a variable exists in each equation, and a symbol of an empty element is displayed at an intersection on a matrix where no variable exists in each equation. Arbitrarily selects a set A of which an equation is to be formed, the set B formed by one or more of the selected sets A and all of the undetermined variables involved has the same requirement. It is determined whether the set A and the set B have the same number of elements. If the set A and the set B have the same number of elements, the undecided variable is symbolically displayed as a deterministic variable, and the equation included in the set A is expressed. Is symbolized as determinate, and in equations other than the set A, the equation including the determinate variable is changed to the symbol of the intersection on the matrix corresponding to the determinate variable and displayed as a symbol indicating involvement. For equations other than the set A, by repeating the process of selecting the set, performing the above-described determination, and performing the above-described determination and display of determination and involvement, all the equations are defined as symbols of determination and involvement. It is configured to be displayed with this. As a result, the constants, variables, and equations containing them are displayed in a list form in an easily viewable manner. In other words, when solving simultaneous equations, it is possible to clearly see how many proof paths are related to each other, and the procedure for finding a solution is easy and has few errors.

[Brief description of the drawings]

FIG. 1 is a diagram showing typical functional divisions of a screen displayed by a method for displaying a solution procedure of a system of simultaneous equations according to the present invention.

2A is a diagram showing a display example when the names of variables written in FIG. 1A are five levels, and FIG. 2B is a diagram showing the names of variables written in FIG. It is a figure showing the example of a display at the time of changing to four steps.

FIG. 3 is a block diagram of a program for displaying a solution procedure of a system of simultaneous equations according to the present invention;

FIGS. 4A and 4B are diagrams for explaining proof of uniqueness of determination. FIG.

FIG. 5 is an electric circuit diagram relating to Example 6.

[Explanation of symbols]

A) Area to enter the name of the equation b) Area to enter the name of the variable c) Area of the flag to specify the relationship between the equation and the variable d) Area of the flag to specify whether the variable is known e) Request for confirmation Area for moving a row Area for moving a row Area for moving a column Area for changing the number of steps in a variable name

Claims (2)

[Claims] 1. A logical procedure comprising display means capable of redrawing and input means capable of inputting a position of two or more dimensions, wherein a logical procedure for deriving the value of each variable of a system of simultaneous equations through interactive input / output is provided. A method of displaying on a screen of a display means a setting of involvement indicating whether or not each equation includes an individual variable, and a pre-simulation of a procedure in which simultaneous calculations are performed on the set equation. And a matrix in which each equation of the system of equations to be processed is arranged in a row and each variable is associated with a column is displayed on the screen of the display means. The symbol of the filling element is displayed at the intersection on the matrix where the variable exists in the matrix, and the symbol of the empty element is displayed at the intersection on the matrix where the variable does not exist in each equation. If a set A formed by the formula is arbitrarily selected, it is determined whether or not one or more of the selected sets A and the set B formed by all of the undetermined variables involved have the same number of elements. If the result of this determination is that set A and set B have the same number of elements, the undecided variables are symbolically displayed as deterministic variables, and the equations contained in the set A are symbolically displayed as deterministic. And, for equations containing the deterministic variables in equations other than the set A, the symbols of the intersections on the matrix corresponding to the deterministic variables are changed to symbols indicating involvement and displayed. In particular, by repeating the process of selecting the set, performing the above-described determination, and performing the above-described determination and display of determination and involvement recursively, all the equations are displayed as symbols of determination and involvement. Solution procedure display method of the system of simultaneous equations to be. 2. A logical procedure comprising a redrawable display means and an input means capable of inputting a position of two or more dimensions, wherein a logical procedure for deriving values of respective variables of a system of simultaneous equations through interactive input / output is provided. A computer-readable recording medium recording a program for displaying a solution procedure of a system of simultaneous equations to be displayed on a screen of a display means, wherein whether or not each equation includes an individual variable is set. It consists of two stages, a pre-simulation of the procedure in which the simultaneous calculation is performed for the set equations, and determination, in which each equation of the system of equations to be processed is made into a row and each variable is made into a column. Is displayed on the screen of the display means, a symbol of a filling element is placed at an intersection on the matrix where a variable exists in each equation, and a matrix where no variable exists in each equation. A procedure for displaying the symbols of the empty elements at the intersections on the box, and when a certain set A made up of the equations is arbitrarily selected by operating the input means, one or more of the selected sets A are selected. And whether or not the set B created by all of the undetermined variables involved has the same number of elements. If the result of this determination is that the sets A and B have the same number of elements, An undecided variable is symbolized as a deterministic variable, an equation included in the set A is symbolized as a determinant, and for an equation other than the set A that includes the deterministic variable, a matrix corresponding to the determinant variable is used. The procedure of changing the symbol of the above intersection to a symbol indicating involvement and displaying it, and for the remaining equations other than the set A, selecting the set, making the determination, and displaying the determination and involvement A computer-readable recording medium storing a program for causing a computer to execute: a procedure of recursively repeating an evaluation process, a procedure of displaying all equations by symbols of determination and involvement by recursively repeating .