KR100533883B1 - Techniques for Graphics Data Processing and Transformation of Linear Expressions, Designed for Graphics Processor - Google Patents

Abstract

The present invention relates to a graphics data processing method and a linear conversion method for efficiently implementing a linear equation in a graphics processor.

Graphics data processing method according to the present invention, if any linear equation E, such as the following equation is input to the graphics processor, the transformation step of finding a transformed linear equation that is the minimum cost and equivalent to the linear equation and ; And a shader coding step of applying a minimum number of vertex instructions to the deformation linear equation.

In addition, the linear conversion method includes a goal setting step of setting a target cost; A matrix transformation step of obtaining any modified linear equation E 'of the linear equation E; A cost calculation step of calculating a cost for calculating the linear equation E and a cost for calculating the modified linear equation E '; Calculating a cost difference value for calculating a difference between a cost required for calculating the modified linear equation E 'and a cost required for calculating the linear equation E; A linear reset step of resetting the modified linear equation E 'to a new linear equation E when the difference in the cost is less than zero or a probability that the difference is less than zero in the process; And a target achievement step of repeating the matrix transformation step or the linear plant reset step until the cost required for the calculation of the linear equation E becomes the target cost.

Description

Graphics Data Processing and Transformation of Linear Expressions, Designed for Graphics Processor

The present invention relates to a graphics data processing method and a linear conversion method in a graphics processor, and more particularly, to a graphics data processing method and a linear conversion method for efficiently implementing a linear expression in a graphics processor.

Graphics processors, which are now commonplace in the graphics world, are evolving so that users can program themselves without having to follow a fixed pipeline. The shader technology of these graphics processors enables various rendering effects to be implemented in real-time and allows processing on a graphics processor (GPU) for a variety of calculations that were only possible on a conventional CPU.

In the existing fixed pipeline, only limited parameters can be obtained because the user can change only parameters. To overcome these shortcomings, efforts have been made to replace each part of the rendering pipeline with user-programmed code. A typical example is the shading language of RenderMan.

This RenderMan shading language is a software rendering system that has the advantage of allowing you to customize the shading and lighting calculations. However, due to its complexity, such a shading scheme is widely used to obtain film or photorealistic images, which are mainly an offline rendering field. Emerging programmable graphics processors provide shaders on hardware to achieve various shading effects in real time.

The main feature of this programmable graphics processor is that it provides vertex shaders and pixel shaders. That is, the programmer can freely implement per-vertex and per-pixel operations. When using a graphics processor, bump mapping, environmental mapping, and reflection mapping, which previously required a lot of CPU calculation, can be performed flexibly on the shader. Therefore, the recent research trend is focused on realizing many advanced rendering techniques that have been performed offline in such a shader, and in particular, the use of pixel shaders with very fast execution time increases rapidly. Doing.

As a technique using a pixel shader, various techniques related to ray tracing and volume rendering have been studied and proposed. However, until now, many studies using pixel shaders have been published, and studies using vertex shaders are still poor.

Meanwhile, linear expression, the simplest form of expression expression, is widely used to express various problems in various fields of engineering as well as computer graphics. In particular, the y = Ax + b linear equation, which corresponds to the affine transform, plays an important role in dealing with various application problems used in computer graphics. In the above linear equations, the cost of matrix operation is high, especially when the size of the matrix is very large and sparse matrix, an efficient calculation method is required.

Until now, such matrix operations have been mainly performed by the CPU, but recently, a method of performing matrix operations using a graphics processor has been developed. Accordingly, the matrix operations can be efficiently implemented on the vertex shader provided by the graphics processor. There is a need to develop a way to optimize shader code.

An object of the present invention devised to meet the above needs is to provide a method for optimizing shader code to efficiently implement an application problem using affine type linear equations on a vertex shader provided by a graphics processor. .

In the graphics data processing method according to the present invention for achieving the above object, if any linear equation E, such as the following equation, is input to the graphics processor, a minimum cost is required for the linear equation and is equivalent to the linear equation. A transformation step of finding a linear equation;

A shader coding step of applying a minimum number of vertex instructions to the deformation linear equation;

Characterized by including.

[Equation]

Here, the linear equation E is a formula for finding a when V is given, where M and A are constant matrices and constant vectors, respectively, and V and a are vectors representing variables.

According to the present invention, there is also provided a computer-readable recording medium having recorded thereon a program for executing a graphics data processing method as described above in a graphics processor.

In addition, a linear conversion method in a prefix processor according to the present invention includes: a target setting step of setting a target cost when an arbitrary linear equation E, such as the following equation, is input to a graphics processor;

A matrix transformation step of obtaining any modified linear equation E 'of the linear equation E;

A cost calculation step of calculating a cost for calculating the linear equation E and a cost for calculating the modified linear equation E ';

Calculating a cost difference value for calculating a difference between a cost required for calculating the modified linear equation E 'and a cost required for calculating the linear equation E;

A linear reset step of resetting the modified linear equation E 'to a new linear equation E when the difference in cost is less than zero or a probability that the difference is less than zero in the process;

A goal achievement step of repeating the matrix transformation step or the linear plant reset step until the cost of calculating the linear equation E becomes the target cost;

Characterized by including.

[Equation]

Here, the linear equation E is a formula for finding a when V is given, where M and A are constant matrices and constant vectors, respectively, and V and a are vectors representing variables.

According to the present invention, there is also provided a computer-readable recording medium having recorded thereon a program for executing a linear conversion method as described above in a graphics processor.

Hereinafter, a graphic data processing method and a linear conversion method in a graphics processor according to an embodiment of the present invention will be described in detail with reference to the accompanying drawings.

Before describing the present invention, a vertex shader to which the present invention is applied will be described first.

The vertex shader is a shader for replacing per-vertex operations before rasterization in the existing real-time graphics pipeline, and its biggest purpose is to perform geometric transformation and lighting calculations. Vertex shaders provide machine language-level Single Instruction Multiple Data (SIMD) instructions, enabling parallel processing. Vertex shader instructions are 4-wide SIMD instructions that perform operations in register units consisting of four fields.

As vertex shader instructions, there are instructions such as MUL, ADD, MAD, DP3, and DP4. MUL is an instruction that performs four multiplications at the same time, ADD is an instruction that performs four additions at the same time, MAD is an instruction that performs four multiplications and four additions at the same time, and DP3 is three multiplications and two additions at the same time. DP4 is a command to perform 4 multiplications and 3 additions at the same time.

In the case of vertex shaders, it is possible to simplify linear operations by using these instructions. NVIDIA GeForce3 & 4 and ATI RADEON8500 and later are commonly used as graphics processors, in which case the number of vertex instructions available in a vertex shader is limited and the number of registers available is limited. Efforts are needed to utilize resources efficiently. Moreover, since the number of instructions and the calculation time are closely related, reducing the number of instructions has a great meaning.

On the other hand, the vertex shader must be able to selectively operate on the four data for linear operation because the four fields of the register are operated at the same time. To do this, the vertex shader provides a swizzling feature that allows you to select or reorder only the four pieces of data in a register. Using this swizzling feature does not incur any performance penalty.

The present invention can be applied to the graphics processor of the vertex shader that supports 4-wide SIMD calculation, supports MOV, ADD, MUL, MAD, DP4 instructions, and supports swizzling.

In general, in the graphics field, affine type linear equations such as Equation 1 occur very frequently. The present invention is intended to represent the most optimized instruction using a vertex shader provided by a programmable graphics processor.

Linear E is a formula for a given V, where M and A are constant matrices and constant vectors, and V and a are vectors representing variables. E in Equation 1 may be represented by a determinant such as Equation 2 below.

In the vertex shader, since the SIMD instruction is calculated in four data units, the linear E is divided into four, and the remainder is filled with '0' to fit the linear E in multiples of four. Then, as shown in Equation 2 above, a determinant in which '0' is added can be obtained. In the above, the size of the matrix M is 4n × 4n (4n = m + α, 0 ≦ α <4).

If the partial 4 × 4 matrices of the matrix M are called M _ij (i, j = 0, 1, 2, ..., n-1), and the equation calculated by the partial 4 × 4 matrix is developed, Same as

Now, we need to convert the previous expression to the SIMD instruction of the vertex program. First, we multiply the submatrix M _ij , which is a 4 × 4 matrix in the entire linear equation, by the four-dimensional variable vector V of the variables, and then Finally, R _i is obtained by adding the resulting S _ij and the constant vector A _i . If this is expressed as an equation, Equation 4 below.

Now, the calculation for linear E is changed to fit a 4-wide SIMD instruction as shown in Equation 5 below. Here, S _ij , A _i , R _i are all four-dimensional vectors.

Equation 5 above is modified to be easy to calculate using the vertex shader 4-wide SIMD instruction. If the matrix M, which is a constant matrix, is a sparse matrix, it is possible to calculate a linear expression with much fewer instructions, ignoring the calculation of multiplication and addition with '0'. Thus, the present invention proposes a method of optimizing shader code to minimize the number of instructions used to calculate a given linear equation.

1 is an operation flowchart showing a graphics data processing method according to the present invention. The graphics data processing method according to the present invention comprises a transform step and a shader coding step. The transform step transforms a given linear equation into another type of equivalent linear equation that is more efficient, and the shader coding step calculates the transformed linear equation using the minimum SIMD instruction.

That is, the transform step is to find a better form of linear equation that is equivalent to a given linear equation and requires fewer instructions, and the calculation step is the most optimized SIMD instruction required to compute the transform equation found in the transform step. Finding them.

First, when the linear E and the final target cost are input to the graphics processor (S11), the target cost is set (S12).

Then, the cost required for the calculation of the linear equation E is calculated (S13). Here, the cost required for the calculation of the linear E means the number of vertex instructions required when the linear E is calculated.

The process of calculating the cost C (E) of the linear equation E will be described. First, as shown in Equation 2 above, the remainder of the linear equation E is filled with '0' so that the magnitude of the linear equation E is a multiple of four, and divided by a 4 × 4 submatrix.

Then, the cost C (E) of the linear equation E is calculated using Equation 6 below.

Where C (E) is the linear cost, Is the smaller of the maximum number of nonzero elements in a row for the (i, j) th submatrix and the number of rows with nonzero elements, Is the number of submatrices in which a maximum of the number of non-zero elements in a row is greater than the number of rows with non-zero elements ( Subtracted 1 from).

The reason why the cost C (E) of the linear E is calculated as in Equation 6 will be described later.

Next, the modified linear equation E 'is obtained by deforming the linear equation E (S14). Since the number of vertex instructions required to implement linear E is closely related to the form of matrix M and vector A of linear E, changing the shape of matrix M and vector A by modifying the given linear E Will change. In the present invention, in order to find a linear equation that requires the least cost, a modified linear equation E 'equivalent to the linear equation E is obtained.

As shown in FIG. 2, the rule for obtaining the modified linear equation E 'is to exchange rows i and j in the matrix M and the vectors a, V, and A for the linear equation E of Equation 1, and The j column is exchanged and obtained.

Next, the cost C (E ') is calculated by applying the modified linear equation E' to Equation 6 (S15). The difference C between the cost C (E ') of the modified linear equation E' obtained in step S15 and the cost C (E) of the linear equation E obtained in step S13 is obtained (S16), and 1 and a probability value Sets a smaller value to p (S17), and returns to step S14 if p is smaller than any real value between 0 and 1, and if p is not less than any real value between 0 and 1, the deformation linear equation E 'Is reset to a new linear E (S19).

That is, if C is less than zero, the probability P is set to 1 because it is larger than 1, and in step S18, since p is unconditionally larger than any real value between 0 and 1, the process proceeds to step S19, and if C is greater than 0, the probability value Since is less than 1 and close to 0 according to C, if any real value between 0 and 1 is selected, p may be greater than or less than that real value, in which case proceed to step S19. If p is not greater than the selected real value, the flow proceeds to step S14.

Summarizing this, if there is a high probability that the cost C (E ') of the deformation linear E' may be less or less than the cost C (E) of the linear E, then set this deformation linear E 'as the new linear E do. However, if the cost C (E ') of the deformation linear E' is more likely to be greater than the cost C (E) of the linear E, a new deformation linear E 'is obtained.

If the cost of the linear E is less than or equal to the target cost (S20) for the newly set linear E (S20), update the target cost (S21), and if the target cost update count is greater than or equal to the maximum number of iterations (S22) Recognize the type E as the least expensive deformation linear equation and calculate it by applying the optimal vertex instruction to the deformation linear equation (S23).

On the other hand, if the cost of the linear equation E is greater than the target cost in step S20 or the target cost update number is smaller than the maximum number of repetitions in step S22, the flow advances to step S14 to obtain a new modified linear equation E '.

Next, the reason why the cost of the linear equation E is calculated as in Equation 6 and the vertex coding step will be described.

In order to code Equation 5 using the minimum SIMD instruction, a partial calculation method corresponding to Process 1 of Equation 4 and a minimum calculation method for the vector sum corresponding to Process 2 should be defined.

First, referring to the partial product, the contents of the multiplication S _ij of the vector V composed of 4 × 4 M _ij and the four variables V are expressed by Equation 7 below.

That is, to calculate S _ij , 16 multiplications and 12 additions are required. However, if '0' is present in the elements of M _ij , the number of operations is reduced. Also, since vertex instructions are SIMD instructions, the total number of instructions is reduced if one instruction performs as many operations as possible. The total number of instructions required for this one calculation is the cost value for that calculation.

In the present invention, a minimum calculation method is proposed according to the shape of M _ij . Here, means the form of the elements other than '0' in M _ij distribution as the shape of M _ij.

In the present invention, the linear equations are calculated by column-major multiplication and row-major multiplication according to the shape of M _ij . 3 is a diagram representing a column-first product method, and FIG. 4 is a diagram representing a row-first product method.

First, referring to FIG. 3, the column-first product method uses a MAD instruction and a MUL instruction to calculate S _ij . S _ij is calculated using one MUL instruction and one C _E (i, j) -1 MAD instruction when the number of nonzero '0' elements in a row is C _E (i, j). Thus, the total number of instructions used is C _E (i, j).

Meanwhile, the row-first multiplication method calculates S _ij by using the DP4, DP3, and MUL instructions as shown in FIG. 4, and refers to a general form of performing a matrix operation using an inner product. When the number of lines with the elements other than '0' in the sub-matrix M _ij be said _E r (i, j), and calculates the S _ij to DP4 of the command r _E (i, j). Thus, the total number of instructions used is r _E (i, j).

Each S _ij may be calculated using the two calculation methods described above. However, to reduce cost, S _ij is calculated by selecting a method that requires fewer instructions. That is, the submatrices with C _E (i, j) <r _E (i, j) are computed in column-first order, and the submatrices with C _E (i, j)> r _E (i, j) are rows. Calculate using the first order method.

Sub-matrix where C _E (i, j) = r _E (i, j) is calculated in a column-first order method for the following reasons. For in the calculation of the line type E, after calculating the S _ij must do more vector sum calculation for adding and A _i using the ADD instruction and, if S _ij and A _i are both zero vector is a vector sum operation This becomes unnecessary. In addition, in the case of the column-first multiplication method, when there is a vector that has already been calculated, if the MAD instruction is used instead of the MUL instruction that is used first, the vector and vector sum operation that have already been calculated are performed together, so that a separate ADD instruction is not added. This can effectively reduce the number of instructions. Therefore, as mentioned above, the submatrices with C _E (i, j) = r _E (i, j) are calculated in a column-first order manner.

That is, the cost required to calculate any partial product S _ij Is a smaller value of C _E (i, j) and r _E (i, j), which is expressed by Equation 8 below.

A partial matrix _{C E (i, j)>} r E (i, j) is a line-first case of the S _ij calculated as the product scheme is to use the ADD instruction again to adding the S _ij and A _i, such Cost for vector sum Is applied only to the submatrices calculated by the row-first-product method, and the expression is expressed by the following equation (9).

here, Is the number of S _ij calculated by the row-first order method.

Therefore, the cost required for the calculation of the linear equation E can be obtained as the sum of the cost required for the vector product and the cost required for the sum of the vectors, which is expressed by Equation 6 above.

The constant vector of the linear equations before (a) and after (b) before applying the present invention to the 4 ^th- order Runge-Kutta method is shown in FIG. 5. The cost before applying the present invention is 80, but the cost after applying the present invention is 56, which allows linear calculation using 120 vertex instructions.

6 shows a linear constant vector before (a) and after (b) application of the present invention to a Gauss-Seidel method. The cost before applying the present invention is 112, but the cost after applying the present invention is reduced to 72.

7 shows the linear constant vector before (a) and after (b) before applying the present invention to the calculation of two-dimensional wave equation. The cost before applying the present invention is 82, but the cost after applying the present invention is reduced to 63.

While the invention has been described above based on the preferred embodiments thereof, these embodiments are intended to illustrate rather than limit the invention. It will be apparent to those skilled in the art that various changes, modifications, or adjustments to the above embodiments can be made without departing from the spirit of the invention. Therefore, the protection scope of the present invention will be limited only by the appended claims, and should be construed as including all such changes, modifications or adjustments.

According to the present invention, when calculating a linear equation using a graphics vertex shader, since the number of instructions can be minimized to be calculated using an optimized SIMD instruction, processing speed is increased and a limited resource can be used more efficiently. There is an advantage.

1 is an operation flowchart showing a graphics data processing method according to the present invention;

2 is a diagram showing a rule for obtaining a deformation linear equation E ';

3 is a diagram representing a column-first product method,

4 is a diagram representing a row-priority method;

FIG. 5 is a diagram showing a linear constant vector before (a) and after (b) application of the method according to the present invention to a 4 ^th -order Runge-Kutta method.

FIG. 6 is a diagram illustrating a constant vector of linear equations before (a) and after (b) application of the method according to the present invention to a Gauss-Seidel method.

7 is a diagram illustrating a linear constant vector before (a) and after (b) of applying the method according to the present invention to the calculation of two-dimensional wave equation.

Claims (16)

delete delete When an arbitrary linear equation E is input to the graphics processor as shown in the following equation, graphics data processing is performed that finds a deformation linear equation that is the minimum cost for the linear equation E and is equivalent to the linear equation E and applies the minimum number of vertex instructions. In the method, A goal setting step of setting, by the graphics processor, a target cost required for the calculation of the linear E; A matrix transformation step of the graphics processor obtaining any modified linear equation E 'equivalent to the linear equation E; A cost calculation step of calculating, by the graphics processor, the cost of the calculation of the linear equation E and the cost of the calculation of the deformation linear equation E '; A cost difference calculation step of calculating, by the graphics processor, a difference between a cost required for the calculation of the modified linear equation E ′ and a cost required for the calculation of the linear equation E ′; A linear reset step of the graphics processor resetting the modified linear equation E 'to a new linear equation E when the cost difference is less than zero or a probability that the cost difference is less than zero in the process; A target achievement step of the graphics processor repeatedly performing the matrix transformation step or the linear plant reset step until the cost of calculating the linear equation E becomes the target cost; And a shader coding step in which the graphics processor applies a minimum number of vertex instructions to a linear equation that requires the least cost obtained as a result of the target achievement step. [Equation] Here, the linear equation E is a formula for finding a when V is given, where M and A are constant matrices and constant vectors, respectively, and V and a are vectors representing variables. The method of claim 3, wherein And repeating the goal setting step or the goal achievement step while changing the target cost until the target number of times of updating the target cost becomes a predetermined value. When an arbitrary linear equation E is input to the graphics processor as shown in the following equation, graphics data processing is performed that finds a deformation linear equation that is the minimum cost for the linear equation E and is equivalent to the linear equation E and applies the minimum number of vertex instructions. In the method, A cost calculation step of calculating, by the graphics processor, the cost for the calculation of the linear E; A matrix transformation step of the graphics processor obtaining all the modified linear equations E ′ of the linear equations E ′; A cost calculation step of the strain line type for calculating, by the graphics processor, the cost of the calculation of all the deformation linear equations E '; A search step of the graphics processor comparing the cost of the calculation of the linear equation E with the cost of the calculation of all the deformation linear equation E ', and finding a deformation linear equation having the minimum cost; And a shader coding step in which the graphics processor applies a minimum number of vertex instructions to a linear equation that requires the least cost resulting from the searching step. [Equation] Here, the linear equation E is a formula for finding a when V is given, where M and A are constant matrices and constant vectors, respectively, and V and a are vectors representing variables. The method according to claim 3 or 5, The cost calculation step, And dividing the linear equation into 4 × 4 submatrices and calculating the result by the following equation. [Equation] Where C (E) is the linear cost, Is the smaller of the maximum number of nonzero elements in a row for the (i, j) th submatrix and the number of rows with nonzero elements, Is the number of submatrices in which a maximum of the number of non-zero elements in a row is greater than the number of rows with non-zero elements ( Subtracted 1 from). The method of claim 6, The cost calculation step, And dividing the remaining portion into '0' so that the linear size is a multiple of four, and dividing it into 4x4 submatrices. The method according to claim 3 or 5, The matrix transformation step, And i rows and j rows in the matrix M and the vectors a, V, and A, and i columns and j columns in the matrix M. The method according to claim 3 or 5, The shader coding step, Split the minimum cost linear equation into 4 × 4 submatrices, For each partitioned sub-matrix, shaders are coded in a column-first order manner when the number of columns with elements other than '0' is less than or equal to the number of rows with elements other than '0' and '0'. The method of processing graphics data according to claim 1, wherein the number of partial matrixes having non-zero elements is larger than the number of rows having non-zero elements is shader coded in a row-first order manner. When an arbitrary linear equation E is input to the graphics processor as shown in the following equation, graphics data processing is performed that finds a deformation linear equation that is the minimum cost for the linear equation E and is equivalent to the linear equation E and applies the minimum number of vertex instructions. A computer-readable recording medium having recorded thereon a program for executing the method, The graphics data processing method, A goal setting step of setting, by the graphics processor, a target cost required for the calculation of the linear E; A matrix transformation step of the graphics processor obtaining any modified linear equation E 'equivalent to the linear equation E; A cost calculation step of calculating, by the graphics processor, the cost of the calculation of the linear equation E and the cost of the calculation of the deformation linear equation E '; A cost difference calculation step of calculating, by the graphics processor, a difference between a cost required for the calculation of the modified linear equation E ′ and a cost required for the calculation of the linear equation E ′; A linear reset step of the graphics processor resetting the modified linear equation E 'to a new linear equation E when the cost difference is less than zero or a probability that the cost difference is less than zero in the process; A target achievement step of the graphics processor repeatedly performing the matrix transformation step or the linear plant reset step until the cost of calculating the linear equation E becomes the target cost; And a shader coding step in which the graphics processor applies a minimum number of vertex instructions to the least costly linear equation obtained as a result of the goal achievement step. [Equation] Here, the linear equation E is a formula for finding a when V is given, where M and A are constant matrices and constant vectors, respectively, and V and a are vectors representing variables. In the linear deformation method for finding a deformation linear equation which is the minimum cost for the linear equation E and is equivalent to the linear equation E, when an arbitrary linear equation E is input to the graphics processor. A goal setting step of setting, by the graphics processor, a target cost required for the calculation of the linear E; A matrix transformation step of the graphics processor obtaining any modified linear equation E 'equivalent to the linear equation E; A cost calculation step of calculating, by the graphics processor, the cost of the calculation of the linear equation E and the cost of the calculation of the deformation linear equation E '; A cost difference calculation step of calculating, by the graphics processor, a difference between a cost required for the calculation of the modified linear equation E ′ and a cost required for the calculation of the linear equation E ′; A linear reset step of the graphics processor resetting the modified linear equation E 'to a new linear equation E when the cost difference is less than zero or a probability that the cost difference is less than zero in the process; And a target achievement step of repeatedly performing the matrix transformation step or the linear plant reset step until the cost of the calculation of the linear equation E becomes the target cost. [Equation] Here, the linear equation E is a formula for finding a when V is given, where M and A are constant matrices and constant vectors, respectively, and V and a are vectors representing variables. The method of claim 11, And repeating the goal setting step or the goal achievement step while updating the target cost until the target cost update number becomes a predetermined value. The method of claim 11, The cost calculation step, And dividing the linear equation into 4 × 4 submatrices and calculating the linear equation by the following equation. [Equation] Where C (E) is the linear cost, Is the smaller of the maximum number of nonzero elements in a row for the (i, j) th submatrix and the number of rows with nonzero elements, Is the number of submatrices in which a maximum of the number of non-zero elements in a row is greater than the number of rows with non-zero elements ( Subtracted 1 from). The method of claim 13, The cost calculation step, And dividing the remaining portion into '0' so that the size of the linear expression is a multiple of four and dividing it into 4x4 submatrices. The method of claim 11, The matrix transformation step, And i rows and j rows in the matrix M and the vectors a, V, and A, and columns i and j in the matrix M are exchanged. When an arbitrary linear equation E is input to the graphics processor as shown in the following equation, a program for executing the linear deformation method that finds the deformation linear equation that is the least cost for the linear equation E and is equivalent to the linear equation E is recorded. In a computer-readable recording medium, The linear deformation method, A goal setting step of setting, by the graphics processor, a target cost required for the calculation of the linear E; A matrix transformation step of the graphics processor obtaining any modified linear equation E 'equivalent to the linear equation E; A cost calculation step of calculating, by the graphics processor, the cost of the calculation of the linear equation E and the cost of the calculation of the deformation linear equation E '; A cost difference calculation step of calculating, by the graphics processor, a difference between a cost required for the calculation of the modified linear equation E ′ and a cost required for the calculation of the linear equation E ′; A linear reset step of the graphics processor resetting the modified linear equation E 'to a new linear equation E when the cost difference is less than zero or a probability that the cost difference is less than zero in the process; And a target achievement step of repeatedly performing the matrix transformation step or the linear plant reset step until the cost of the calculation of the linear equation E becomes the target cost. media. [Equation] Here, the linear equation E is a formula for finding a when V is given, where M and A are constant matrices and constant vectors, respectively, and V and a are vectors representing variables.