A method for accelerating a local path planning algorithm of a mobile robot

Abstract

The application discloses a kind of acceleration method of local path planning algorithm of mobile robot, comprising: 1) the optimization thought based on graph optimization, corresponding hypergraph of each constraint condition in time elastic band method TEB is constructed, and the new form of TEB optimization problem is established;2) design iteration solving form under the alternating direction multiplier method ADMM framework, including iteration solving formula and termination condition, according to the corresponding result of hypergraph calculation, the new form of TEB optimization problem obtained is solved;3) design multi-step conjugate gradient solving method, responsible for solving part of variables in iteration solving form under the ADMM framework, replace part of iteration solving formula of original iteration solving form by approximating optimal solution along a set of conjugate gradient direction, to accelerate ADMM iteration solving to obtain optimal solution.The application solves optimization problem in local path planning algorithm by acceleration, alleviate the problem that local path planning algorithm occupies many resources, time-consuming and so on in complex scene, improve the speed of solving optimization problem.

Description

Technical Field

[0001] This invention relates to the technical fields of local path planning for mobile robots, two-wheeled differential drive robots, and convex optimization, and in particular to a method for accelerating local path planning algorithms for mobile robots. Background Technology

[0002] With the development of technology, mobile robots have become a key research area in intelligent manufacturing and services. One of the core issues in the field of mobile robots is path planning. The dynamic uncertainty and complexity of the working environment in which robots operate require them to have excellent autonomous obstacle avoidance capabilities. Therefore, it is crucial to enable robots to plan paths efficiently and safely in complex and dynamic environments to achieve autonomous navigation and task execution.

[0003] Path planning research is crucial for improving the autonomy, adaptability, and intelligence of robots. With continuous technological advancements, an increasing number of path planning methods have been proposed and studied to meet the needs of various application scenarios. Path planning can generally be divided into global path planning and local path planning. Local path planning, a key problem in robot navigation, involves planning a path for the robot to avoid obstacles and adhere to safety and feasibility constraints within a known map or perceived environment in a short time. Common local path planning methods include Dynamic Window (DWA), Temporal Elastic Band (TEB), and Artificial Potential Field (APF) methods. Among these methods, DWA and APF are simple and fast but prone to getting trapped in local minima. TEB plans better and smoother paths that can satisfy different vehicle models, but its high computational complexity affects real-time performance. Therefore, TEB is more commonly used in practical applications to obtain optimal local paths. To address the computational complexity of TEB, graph optimization is employed as a feasible solution.

[0004] Graph optimization, by modeling multi-objective optimization problems of sparse models in TEB using hypergraphs, simplifies model representation, reduces computational complexity, and improves computational efficiency, thereby reducing the computational load and storage resource consumption of TEB. Compared to traditional graphs, hypergraphs offer a simpler structure and significantly lower costs, while also accelerating computation through parallel processing of multiple unrelated nodes and constraint edges. Some studies have shown that this method is highly flexible and adaptable to different robot kinematics and application requirements. However, while graph optimization simplifies model construction and data storage, the algorithmic complexity of solving the linear equations of the sparse coefficient matrix in the optimization problem remains high. This means that in complex scenarios, excessive computation can negatively impact the real-time performance of TEB.

[0005] TEB transforms the optimization objective of the entire system into a quadratic programming problem through graph optimization. Solving this convex optimization problem is also a major factor affecting the algorithm's performance. Solving a quadratic programming problem can be reduced to solving a system of linear equations, where the coefficient matrix is ​​sparse, symmetric, and positive definite. Common methods in the g2o framework for solving this problem include the LM method and Cholesky decomposition. While gradient descent can approach the optimal solution relatively quickly, the matrix inversion operation involved has high computational complexity. The idea of ​​decomposing the coefficient matrix can be accelerated by designing parallel schemes to process unrelated columns in parallel, but the algorithm itself has high complexity, and the parallelization effect is not significant in small-scale problems.

[0006] There is a wealth of research on solving optimization problems, among which the Alternating Direction Multiplier Method (ADMM) is commonly used for solving multi-objective optimization problems. This method decouples the relationships between multiple variables to update them in parallel, and can converge to the optimal solution or its neighborhood relatively quickly. However, in the original method, the analytical solution for updating the variables often needs to be calculated by matrix inversion during the iterative update process. This complexity of inversion can affect the efficiency of the algorithm when solving large-scale problems. Summary of the Invention

[0007] The purpose of this invention is to accelerate the solution of optimization problems in local path planning algorithms, thereby alleviating the problems of high resource consumption and high time consumption in complex scenarios. A method for accelerating local path planning algorithms for mobile robots is proposed. The method uses ADMM to solve the TEB optimization problem and uses a designed multi-step conjugate gradient solution method to iteratively solve the analytical solutions of each variable in ADMM in each step update, reducing the complex calculations caused by matrix inversion and improving the speed of solving optimization problems.

[0008] To achieve the above objectives, the technical solution provided by this invention is: a method for accelerating a local path planning algorithm for a mobile robot, comprising the following steps:

[0009] 1) Based on the optimization approach of graph optimization, a hypergraph corresponding to each constraint condition in the Time Elastic Band Method (TEB) is constructed, and a new form of the TEB optimization problem is established.

[0010] 2) Design an iterative solution form under the Alternating Direction Multiplier Method (ADMM) framework, including iterative solution formulas and termination conditions, and solve the new form of the TEB optimization problem based on the corresponding results of the hypergraph calculation;

[0011] 3) Design a multi-step conjugate gradient solution method to solve some variables in the iterative solution form under the ADMM framework in step 2) in parallel. By approximating the optimal solution along a set of conjugate gradient directions, it replaces some iterative solution formulas in the original iterative solution form, thereby accelerating the ADMM iterative solution to obtain the optimal solution.

[0012] Furthermore, in step 1), the path points and map information obtained by TEB are used as vertices, and the constraints involving these vertices are used as edges to construct the corresponding hypergraph.

[0013] Furthermore, in step 1), the new form of the TEB optimization problem is as follows:

[0014] TEB describes path planning as a multi-objective optimization problem based on the constraints of mobile robots:

[0015]

[0016] In the formula, x * Let B be the optimal solution to the current problem, and let f be the combined state of the mobile robot's pose and time series. kt Let σ be the objective function corresponding to the kt-th constraint. kt The weights are the corresponding weights of the objective function for the kt-th constraint.

[0017] Based on the optimization approach of graph optimization, the TEB optimization problem can be viewed as:

[0018] Δx * =argmin Δx F(x+Δx)=argmin Δx (Δx T HΔx+bΔx)

[0019] That is, a quadratic programming problem:

[0020] minΔx T HΔx+bΔx

[0021] stΔx∈C

[0022] In the formula, Δx * It is the current pose x and the optimal solution x * The difference; F(x+Δx) is the overall objective function in the TEB optimization problem when the current pose x changes by Δx; C is the range of Δx determined by the constraints, which is a convex set; H is the Hessian matrix of the error function of each constraint corresponding to the current pose x, b is the coefficient of the first-order term, and we have:

[0023]

[0024] In the formula, J is the Jacobi matrix corresponding to the current pose x, representing the error function of each constraint. kt Ω is the Jacobi matrix corresponding to the current pose x for the error function of the kt-th constraint, and Ω is the total information matrix. kt Let be the information matrix corresponding to the kt-th constraint. The information matrix is ​​used to represent the weight of each constraint and the correlation parameters between different poses.

[0025] To simplify the representation of the problem, let h(Δx) = Δx T HΔx+bΔx is used to represent the function of interest in the original quadratic programming problem. At the same time, an indicator function g(z) under a convex set C is added, where z is an auxiliary variable used to obtain the iterative solution formula below. Through these two steps, the original quadratic programming problem is transformed into the following new form:

[0026] minh(Δx)+g(z)

[0027] stΔx-z=0,Δx∈C

[0028] in,

[0029]

[0030] Furthermore, in step 2), the iterative solution form under the ADMM framework includes the iterative solution formula and termination condition, as detailed below:

[0031] The augmented Lagrangian function of a new form of the TEB optimization problem is:

[0032]

[0033] To simplify the formula, let r = Δx - z, u = y T / ρ, which transforms the augmented Lagrangian function into the following form:

[0034]

[0035] In the formula, y T Here, ρ is the Lagrange multiplier, r is the residual, and u is the substitution variable used to simplify the formula.

[0036] Therefore, according to ADMM, the corresponding iterative solution formula can be obtained as follows:

[0037] x t+1 =(H+ρI) -1 (ρ(z t -u t )-b)

[0038] z t+1 =ΠC (x t+1 +u t )

[0039] u t+1 =u t +x t+1 -z t+1

[0040] In the formula, x t z t u t x represents the update value of ADMM in the t-th iteration. t+1 z t+1 u t+1 These are the update values ​​for the (t+1)th iteration of ADMM, where I is the identity matrix and Π C (x t+1 +u t ) means to find (x) t+1 +u t The projection of ) onto the convex set C;

[0041] The corresponding termination condition for iterative updates is:

[0042]

[0043] In the formula, r t Let s be the residual of the t-th iteration. t Let z be the dual residual of the t-th iteration. t-1 Let z be the updated value of ADMM in the (t-1)th iteration, and n be the dimension of matrix H; ε abs It is the absolute difference, ε rel It is a relative difference and needs to be set according to the specific requirements of the problem.

[0044] Furthermore, in step 2), the corresponding results of the hypergraph calculation include the Jacobi matrix and Hessian matrix corresponding to the current pose, calculated by the hypergraph according to the objective function of the constraint conditions.

[0045] Furthermore, in step 3), the multi-step conjugate gradient solution method is as follows:

[0046] In the ADMM iterative solution formula, x t+1 The update step is viewed as solving the linear system of equations (H+ρI)x t+1 =ρ(z) t -u t The solution to )-b is given here by denoting the coefficient matrix (H+ρI) as G, and the constant term ρ(z) as b. t -u tLet )-b be denoted as b_new, the required number of parallel steps be denoted as p, and the threshold be denoted as ε. The multi-step conjugate gradient solution method performs the following operations: with (x t Using G, b_new, p, ε as input, solve for x. t+1 The value is calculated until the gradient meets the threshold condition, and this operation replaces the matrix inversion operation in the ADMM iterative solution formula. The specific steps are as follows:

[0047] 3-1) Setting initial values: Set the initial value of the gradient in the k-th iteration to... The conjugate direction is set to d. k,1 =-g k Where k is the number of iterations in the multi-step conjugate gradient solution method and In the k-th iteration, x represents t The updated value, d k,i Let i represent the conjugate direction vector of the i-th step in the k-th iteration, where i = 1, 2, ..., p;

[0048] 3-2) Convergence detection: If ||g k If || < ε, then stop iterating and outputting. As x t+1 Update the value; otherwise, continue iterating.

[0049] 3-3) Calculation:

[0050] iter k,1 =Gd k,1

[0051]

[0052]

[0053] In the formula, iter k,i α represents the result of the matrix-vector multiplication calculation required for the i-th update in the k-th iteration. k,i Let β be the step size needed for the i-th update in the k-th iteration. k,i These are the combination coefficients needed for the i-th update in the k-th iteration;

[0054] 3-4) If p > 1, then calculate sequentially:

[0055]

[0056] iter k,i =Gd k,i

[0057]

[0058]

[0059] Since the first step of the update has already been performed, i = 2, 3, ..., p;

[0060] 3-5) Calculate the update value for this iteration:

[0061]

[0062]

[0063]

[0064] 3-6) Let k = k + 1, then go to step 3-2.

[0065] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0066] 1. This invention establishes a new form of the TEB optimization problem and uses the ADMM distributed optimization framework to accelerate the solution of the new form of the problem, thereby reducing the computation time of TEB when processing large amounts of data in complex environments, which is conducive to TEB adapting to the real-time requirements of various scenarios.

[0067] 2. This invention solves some variables of the ADMM iterative solution formula by using a multi-step conjugate gradient solution method. It replaces the matrix inversion calculation in the original formula with the method of approximating the optimal solution along a set of conjugate gradient directions. Generally, the optimal solution can be obtained through multiple iterations. At the same time, only a few matrix-vector multiplications involving sparse matrices are required in each iteration, which can greatly reduce the computational complexity of ADMM. Compared with the method used in graph optimization to solve sparse linear equation systems, it also has a faster convergence speed.

[0068] 3. This invention improves the conjugate gradient method by calculating multiple update values ​​in one step. Compared with the original method, which requires 2k matrix-vector multiplications when the number of convergence steps is k, the improved multi-step conjugate gradient solution method reduces the number of convergence steps to k / p and only requires k matrix-vector multiplications in total when the number of parallel steps is p. The parallel scheme reduces the amount of computation and the number of iterations. Attached Figure Description

[0069] Figure 1 This is an overall flowchart of the method of the present invention.

[0070] Figure 2 This is a structural diagram of the ADMM framework of the method of the present invention.

[0071] Figure 3 This is a flowchart of the multi-step conjugate gradient solution method of the present invention. Detailed Implementation

[0072] The present invention will be further described below with reference to specific embodiments.

[0073] This embodiment provides a method for accelerating a local path planning algorithm for a mobile robot, such as... Figure 1 As shown, this includes constructing a hypergraph, establishing a new form of the TEB optimization problem, designing an iterative solution form under the ADMM framework, calculating the corresponding results based on the hypergraph, and designing a multi-step conjugate gradient solution method, including the following steps:

[0074] 1) Based on the optimization approach of graph optimization, construct a hypergraph corresponding to each constraint in TEB and establish a new form of TEB optimization problem.

[0075] The construction of the hypergraph corresponding to each constraint in TEB specifically involves using n partial global path points obtained by TEB for local path planning and some obstacle points on the map as vertices, and using the velocity constraints, dynamic constraints, path constraints, time constraints, etc. of these vertices as edges to construct the corresponding hypergraph. According to the default parameter settings of TEB, the number of partial global path points obtained is n = 64 or 66.

[0076] The new form of the TEB optimization problem is as follows:

[0077] TEB describes path planning as a multi-objective optimization problem based on the constraints of mobile robots:

[0078]

[0079] In the formula, x * Let B be the optimal solution to the current problem, and let f be the combined state of the mobile robot's pose and time series. kt Let σ be the objective function corresponding to the kt-th constraint. kt The weights are the corresponding weights of the objective function for the kt-th constraint.

[0080] Based on the optimization approach of graph optimization, the TEB optimization problem can be viewed as:

[0081] Δx * =argmin Δx F(x+Δx)=argmin Δx (Δx T HΔx+bΔx)

[0082] That is, a quadratic programming problem:

[0083] minΔx T HΔx+bΔx

[0084] stΔx∈C

[0085] In the formula, Δx* It is the current pose x and the optimal solution x * The difference; F(x+Δx) is the overall objective function in the TEB optimization problem when the current pose x changes by Δx; C is the range of Δx determined by the constraints, which is a convex set; H is the Hessian matrix of the error function of each constraint corresponding to the current pose x, b is the coefficient of the first-order term, and we have:

[0086]

[0087] In the formula, J is the Jacobi matrix corresponding to the current pose x, representing the error function of each constraint. kt Ω is the Jacobi matrix corresponding to the current pose x for the error function of the kt-th constraint, and Ω is the total information matrix. kt Let be the information matrix corresponding to the kt-th constraint. The information matrix is ​​used to represent the weight of each constraint and the correlation parameters between different poses.

[0088] To simplify the representation of the problem, let h(Δx) = Δx T HΔx+bΔx is used to represent the function of interest in the original quadratic programming problem. At the same time, an indicator function g(z) under a convex set C is added, where z is an auxiliary variable used to obtain the iterative solution formula below. Through these two steps, the original quadratic programming problem is transformed into the following new form:

[0089] minh(Δx)+g(z)

[0090] stΔx-z=0,Δx∈C

[0091] in,

[0092]

[0093] 2) Design an iterative solution form under the ADMM framework, including iterative solution formulas and termination conditions, and solve the new form of the TEB optimization problem based on the corresponding results of the hypergraph calculation.

[0094] The iterative solution form under the ADMM framework, such as Figure 2 As shown, this includes the iterative solution formula and termination condition, as detailed below:

[0095] The augmented Lagrangian function of a new form of the TEB optimization problem is:

[0096]

[0097] To simplify the formula, let r = Δx - z, u = y T / ρ, which transforms the augmented Lagrangian function into the following form:

[0098]

[0099] In the formula, y T Here, ρ is the Lagrange multiplier, r is the residual, and u is the substitution variable used to simplify the formula.

[0100] Therefore, according to ADMM, the corresponding iterative solution formula can be obtained as follows:

[0101] x t+1 =(H+ρI) -1 (ρ(z t -u t )-b)

[0102] z t+1 =Π C (x t+1 +u t )

[0103] u t+1 =u t +x t+1 -z t+1

[0104] In the formula, x t z t u t x represents the update value of ADMM in the t-th iteration. t+1 z t+1 u t+1 These are the update values ​​for the (t+1)th iteration of ADMM, where I is the identity matrix and Π C (x t+1 +u t ) means to find (x) t+1 +u t The projection of ) onto the convex set C;

[0105] The corresponding termination condition for iterative updates is:

[0106]

[0107] In the formula, r t Let s be the residual of the t-th iteration. t Let z be the dual residual of the t-th iteration. t-1 Let z be the updated value of ADMM in the (t-1)th iteration, and n be the dimension of matrix H; ε abs =10 -2 It is the absolute difference, ε rel =10 -4 It is relatively poor.

[0108] The corresponding results of the hypergraph computation include the Jacobi matrix and Hessian matrix corresponding to the current pose, calculated by the hypergraph according to the objective function of the constraint conditions.

[0109] 3) Design a multi-step conjugate gradient solution method to solve some variables in the iterative solution form under the ADMM framework in step 2) in parallel. By approximating the optimal solution along a set of conjugate gradient directions, it replaces some iterative solution formulas in the original iterative solution form, thereby accelerating the ADMM iterative solution to obtain the optimal solution.

[0110] The multi-step conjugate gradient solution method, such as Figure 3 As shown, the specific situation is as follows:

[0111] In the ADMM iterative solution formula, x t+1 The update step is viewed as solving the linear system of equations (H+ρI)x t+1 =ρ(z) t -u t The solution to )-b is given here by denoting the coefficient matrix (H+ρI) as G, and the constant term ρ(u t -z t Let b be denoted as b_new, and set the required number of parallel steps to p = 3, with the threshold set to ε = 10. -3 The multi-step conjugate gradient solution method performs the following operations: with (x t Using G, b_new, p, ε as input, solve for x. t+1 The value is calculated until the gradient meets the threshold condition, and this operation replaces the matrix inversion operation in the ADMM iterative solution formula. The specific steps are as follows:

[0112] 3-1) Setting initial values: Set the initial value of the gradient in the k-th iteration to... The conjugate direction is set to d. k,1 =-g k Where k is the number of iterations in the multi-step conjugate gradient solution method and In the k-th iteration, x represents t The updated value, d k,i Let i represent the conjugate direction vector of the i-th step in the k-th iteration, where i = 1, 2, ..., p;

[0113] 3-2) Convergence detection: If ||g k If || < ε, then stop iterating and outputting. As x t+1 Update the value; otherwise, continue iterating.

[0114] 3-3) Calculation:

[0115] iter k,1 =Gd k,1

[0116]

[0117]

[0118] In the formula, iter k,i α represents the result of the matrix-vector multiplication calculation required for the i-th update in the k-th iteration. k,i Let β be the step size needed for the i-th update in the k-th iteration. k,i These are the combination coefficients needed for the i-th update in the k-th iteration;

[0119] 3-4) If p > 1, then calculate sequentially:

[0120]

[0121] iter k,i =Gd k,i

[0122]

[0123]

[0124] Since the first step of the update has already been performed, i = 2, 3, ..., p;

[0125] 3-5) Calculate the update value for this iteration:

[0126]

[0127]

[0128]

[0129] 3-6) Let k = k + 1, then go to step 3-2.

[0130] In this embodiment, a multi-step conjugate gradient solution method is used to accelerate the ADMM iterative solution. Compared with the Cholesky decomposition method for decomposing sparse symmetric positive definite matrices, the time complexity is O(n^2). 3 The multi-step conjugate gradient method requires only about 22 iterations and 66 matrix-vector multiplications to obtain the final solution. For a sparse matrix H with non-zero elements comprising about 15%, the time complexity is O(0.15n). 3 It has a faster convergence speed.

[0131] In summary, this invention can reduce the computation time of TEB when processing large amounts of data in complex environments, which is beneficial for TEB to adapt to the real-time requirements of various scenarios. At the same time, the improved parallel scheme effectively reduces the computational load of acceleration methods and improves the speed of solving corresponding optimization problems. It has broad research and practical application value and is worth promoting.

[0132] The above-described embodiments are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Therefore, any changes made in accordance with the shape and principle of the present invention should be covered within the protection scope of the present invention.

Claims

1. A method for accelerating a local path planning algorithm for a mobile robot, characterized in that, Includes the following steps: 1) Based on the optimization approach of graph optimization, a hypergraph corresponding to each constraint condition in the Time Elastic Band Method (TEB) is constructed, and a new form of the TEB optimization problem is established. The new form of the TEB optimization problem includes: TEB describes path planning as a multi-objective optimization problem based on the constraints of mobile robots: ； In the formula, Let B be the optimal solution to the current problem, representing the combined state of the mobile robot's pose and time series. Let kt be the objective function corresponding to the kt-th constraint. The weights are the corresponding weights of the objective function for the kt-th constraint. Based on the graph optimization approach, the TEB optimization problem is: ； That is, a quadratic programming problem: ； In the formula, Current pose and the optimal solution The difference; It is the current pose in the TEB optimization problem. Changes The overall objective function; Determined by constraints The range of values ​​is a convex set; The error function of each constraint corresponds to the current pose. The Hessian matrix, Let be the coefficient of the linear term, and we have: ； In the formula, The error function of each constraint corresponds to the current pose. Jacobi matrix, The error function of the kt-th constraint corresponds to the current pose. Jacobi matrix, For the overall information matrix, Let be the information matrix corresponding to the kt-th constraint. The information matrix is ​​used to represent the weight of each constraint and the correlation parameters between different poses. set up This is used to represent the function of interest in the original quadratic programming problem, while also adding a convex set. Indicator functions below , These are auxiliary variables used to obtain the iterative solution formula below. Through these two steps, the original quadratic programming problem is transformed into the following new form: ； in, ； 2) Design an iterative solution form under the Alternating Direction Multiplier Method (ADMM) framework, including iterative solution formulas and termination conditions, and solve the new form of the TEB optimization problem based on the corresponding results of the hypergraph calculation; The iterative solution forms under the ADMM framework include: The augmented Lagrangian function of a new form of the TEB optimization problem is: ； make , This transforms the augmented Lagrange function into the following form: ； In the formula, For Lagrange multipliers, The default penalty parameters are... For residuals; The corresponding iterative solution formula obtained from ADMM is as follows: ； ； ； In the formula, Let be the update values ​​for the t-th iteration of ADMM, respectively. These are the update values ​​for the (t+1)th iteration of ADMM, It is the identity matrix. Expressing the request In convex set Projection on; The corresponding termination condition for iterative updates is: ； In the formula, Let be the residual of the t-th iteration. Let be the dual residual of the t-th iteration. For the (t-1)th iteration of ADMM The updated value, where n is the matrix. The dimension; It's absolutely terrible. It is a relative difference; 3) Design a multi-step conjugate gradient solution method to solve some variables in the iterative solution form under the ADMM framework in step 2) in parallel. This method replaces some of the original iterative solution formulas by approximating the optimal solution along a set of conjugate gradient directions, thereby accelerating the ADMM iterative solution to obtain the optimal solution. Specifically, the ADMM iterative solution formulas... The update step is viewed as solving a system of linear equations. The solution, here the coefficient matrix Recorded as constant term Recorded as Let p be the number of parallel steps required, and let the threshold be denoted as p. The multi-step conjugate gradient solution method performs the following operations: ( , , p Using as input, solve The value of is used until the gradient satisfies the threshold condition, and this operation replaces the matrix inversion operation in the ADMM iterative solution formula.

2. The method for accelerating a local path planning algorithm for a mobile robot according to claim 1, characterized in that, In step 1), the path points and map information obtained by TEB are used as vertices, and the constraints involving these vertices are used as edges to construct the corresponding hypergraph.

3. The method for accelerating a local path planning algorithm for a mobile robot according to claim 2, characterized in that, In step 2), the corresponding results of the hypergraph calculation include the Jacobi matrix and Hessian matrix corresponding to the current pose, which are calculated by the hypergraph according to the objective function of the constraint conditions.

4. The method for accelerating a local path planning algorithm for a mobile robot according to claim 3, characterized in that, In step 3), the multi-step conjugate gradient solution method comprises the following steps: 3-1) Setting initial values: Set the initial value of the gradient in the k-th iteration to... The conjugate direction is set to Where k is the number of iterations in the multi-step conjugate gradient solution method and , In the k-th iteration The updated value, Let represent the conjugate direction vector of the i-th step in the k-th iteration, and ; 3-2) Convergence detection: If Then stop iterating and outputting. As Update the value; otherwise, continue iterating. 3-3) Calculation: ； ； ； In the formula, This represents the result of the matrix-vector multiplication calculation required for the i-th update in the k-th iteration. Let be the step size needed for the i-th update in the k-th iteration. These are the combination coefficients needed for the i-th update in the k-th iteration; 3-4) If Then calculate in sequence: ； ； ； ； Since the first step of the update has already been completed, therefore, here... ; 3-5) Calculate the update value for this iteration: ； ； ； 3-6) Order Proceed to step 3-2).

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