CN112888064B - Improved wireless sensor network DV-HOP positioning optimization method based on M-CG - Google Patents
Improved wireless sensor network DV-HOP positioning optimization method based on M-CG Download PDFInfo
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Abstract
The invention discloses an improved wireless sensor network DV-HOP positioning optimization method based on M-CG, which comprises the following steps: s1, initializing communication radiuses of nodes in a network, finishing random network topology deployment of beacon nodes and unknown nodes, and broadcasting self information and hop values of all beacon nodes to surrounding neighbor nodes; s2, an unknown node in the network receives the information packet broadcast by the neighbor node and forwards the information packet to enable all nodes in the network to obtain the original minimum hop count from the reference beacon node; s3, calculating the new minimum hop count among all beacon nodes; calculating a new minimum hop count from the unknown node to the beacon node; s4, calculating the average hop distance, and optimizing the average hop distance by using the normalization of the new minimum hop value among all nodes as a weight; s5, calculating the distance between each unknown node and the beacon node; s6, calculating the estimated coordinates of the unknown nodes; and S7, iterative refinement. The method improves the robustness and the positioning accuracy of the algorithm.
Description
Technical Field
The invention relates to an improved wireless sensor network DV-HOP positioning optimization method based on M-CG.
Background
Rapid progress of science and computer network technology, wireless network communication technology, sensor technology and the like accelerates the growth of wireless sensor network technology. In the Wireless Sensor Network technology (WSN for short), a large number of miniature irregular sensors are arranged in an area to be monitored, and then data detected by the sensors are transmitted to a related information processing center through a Wireless communication Network, thereby forming a multi-hop self-organizing Network. As one of the four major technologies in the world, WSN will continuously raise new wave of every business. With the rapid development of the Internet of Things (Internet of Things, ioT for short), the WSN technology becomes a crucial technology in the Internet of Things due to its characteristics of reliability, self-organization, fault tolerance, and rapid deployment. The development of WSNs is constantly bringing qualitative changes to our lives. At present, the method is applied to multiple fields of military reconnaissance, monitoring and prevention of natural disasters, agricultural production, medical health, environmental information monitoring and the like, reduces manpower and material resources, greatly improves the rapidity and comprehensiveness of information acquisition, and overcomes the defects of difficulty, untimely time and the like of information acquisition in the traditional technology.
In actual life, WSNs need to know the position of a node, for example, when a forest fire occurs, the source of the fire needs to be quickly positioned; when geological disasters such as earthquake, cave-in collapse and the like occur, in order to evacuate personnel quickly and reduce the loss of manpower, material resources and financial resources, a disaster center needs to be positioned quickly. Therefore, research on node location technology in WSNs is very important. If the WSN can only acquire massive data, the specific positions of the data cannot be known, even if the data are known, no effect can be achieved, and the data with unknown positions are used for research in vain and are meaningless. If the positioning technology can be realized, the target behavior can be predicted, the target can be quickly positioned, the data is analyzed, and corresponding prediction is carried out according to the trend of the data. Therefore, the node positioning technology is always the research focus in the WSN, and is the basis and premise of all practical applications.
Although the wireless sensor network DV-HOP algorithm has many advantages, the DV-HOP algorithm locates unknown nodes through distance vector information and network connectivity, does not directly measure the distance between the nodes, and has a certain influence on the accuracy of location. In the traditional DV-Hop algorithm or the improved DV-Hop algorithm, nodes are generally distributed randomly, and if the difference between a propagation path and a straight line between the two nodes is large and the Hop count between the nodes is more, the error generated by estimating the distance is larger. 2. In the second stage, the same hop distance is used when the distance between the unknown node and the beacon node is estimated, the single average hop distance is adopted to estimate the distance, the real situation of the network cannot be reflected, and the positioning accuracy is influenced.
Disclosure of Invention
The invention provides an improved wireless sensor network DV-HOP positioning optimization method based on M-CG, wherein a new HOP count corresponding to a straight line distance between beacon nodes is calculated by taking a Manhattan distance and an Euclidean distance in combination with an original HOP count path distance between two nodes as a reference, and a CG algorithm is used for carrying out iterative refinement on estimated coordinates of unknown nodes of the beacon nodes, so that the robustness and the positioning accuracy of the algorithm are improved.
In order to achieve the purpose of the invention, the invention adopts the following technical scheme:
the invention provides a DV-HOP positioning optimization method of a wireless sensor network improved based on M-CG, which is characterized by comprising the following steps:
s1, initializing communication radiuses of nodes in a network, finishing random network topology deployment of beacon nodes and unknown nodes, and broadcasting self information and hop values of all beacon nodes to surrounding neighbor nodes;
s2, an unknown node in the network receives the information packet broadcast by the neighbor node and forwards the information packet to enable all nodes in the network to obtain the original minimum hop count from the reference beacon node;
s3, calculating a new minimum hop count between each beacon node by adopting the mean value of the Manhattan distance and the Euclidean distance; calculating a new minimum hop count from the unknown node to the beacon node;
s4, calculating the average hop distance, and optimizing the average hop distance by using the normalization of the new minimum hop value among all nodes as a weight;
s5, calculating the distance between each unknown node and the beacon node according to the product of the new minimum hop count between the unknown nodes obtained in the S3 and the average hop distance obtained in the S4;
s6, calculating the estimated coordinates of the unknown nodes by using a maximum likelihood estimation method;
and S7, carrying out iterative refinement on the estimated coordinates of the unknown nodes by using a CG algorithm.
As a further improvement of the invention: the step S1 includes:
s11, initializing the communication radius of the nodes in the network, and finishing network topology deployment of the beacon nodes and unknown nodes;
s12, broadcasting a data packet containing self number, position information and hop value by the beacon nodes in the wireless sensor network in a flooding mode, wherein the initial value of each beacon node hop value is set to be zero.
As a further improvement of the invention: the step S2 includes:
s21, after receiving the information packet of the beacon node, the neighbor node of the beacon node adds one to the hop count value in the information packet and forwards the hop count value to the neighbor node of the next hop;
s22, the node receiving the information packet selects and stores the information packet with the minimum hop value from a plurality of information packets with the same beacon node number received by the node; the node adds one to the hop count value in the selected information packet and forwards the hop count value to the neighbor node of the next hop; step S22 is repeated until flooding is completed.
As a further improvement of the invention: the step S3 includes:
s31, according to the step S1, because each node in the network is randomly distributed, the hop number of the node between two points is selected randomly, the route between the two points is not a straight line, and the introduced Manhattan distance | x between the beacon nodes is adopted i -x j |+|y i -y j I and Euclidean distance
If the mean value of the two nodes is taken as the curve path distance which truly reflects the original minimum hop count, a new minimum hop count value of a straight line path between the two nodes is correspondingly calculated, and the method specifically comprises the following steps:
wherein the content of the first and second substances,
representing the original minimum number of hops between beacon i and beacon j,
setting a new minimum hop number between a beacon node i and a beacon node j, wherein i, j belongs to B, i is not equal to j, and B is the number of unknown nodes; (x) i ,y i ) And (x) j ,y j ) Coordinates of a beacon node i and a beacon node j are respectively represented;
calculating the new minimum hop count between the beacon node i and the beacon node j
Comprises the following steps:
s32, correcting the original hop count between the unknown node e and the beacon node i to obtain a new minimum hop count between the unknown node e and the beacon node i, which specifically comprises the following steps:
wherein e is an unknown node to be positioned,
for the original minimum number of hops between unknown node e and beacon i,
is the new minimum number of hops between the unknown node e and the beacon node i.
As a further improvement of the invention: the step S4 includes:
s41, calculating the average hop distance of the beacon nodes according to the new minimum hop count obtained among the beacon nodes, specifically:
wherein, hopsize i Representing the average per-hop distance of the beacon i,
representing a new minimum hop count between the beacon i and the beacon j;
s42, in order to enable the average hop distance of the positioning node to be closer to the actual average hop distance, normalizing the new minimum hop value obtained in S3 by each beacon node, wherein the specific formula is as follows:
b is the number of the beacon nodes;
s43, optimizing the delta serving as a weight to obtain a new average hop distance, which specifically comprises the following steps:
as a further improvement of the invention: in step S5, the distance between each unknown node and the beacon node is calculated;
s51, a new minimum hop count of the unknown node e and the beacon node i can be obtained through S32, and the product of the minimum hop count and the average hop distance obtained through S43 is calculated to obtain the distance between each unknown node and the beacon node;
as a further improvement of the invention: in step S6, the coordinates of the unknown node are calculated by the maximum likelihood estimation method:
s61, calculating an estimated coordinate of the unknown node by using a maximum likelihood estimation method; let the coordinate of unknown node e in the network be (x) e ,y e ) (ii) a Let the coordinate of beacon i be (x) i ,y i ) (ii) a According to unknown nodese and beacon i by estimating the distance d ei Calculating the estimated coordinates of the unknown nodes by using a following equation set, wherein the specific equation set comprises:
as a further improvement of the invention: the step S7 includes:
s71, because an error zeta exists between the distance between the unknown node and the beacon node, in order to minimize the error, an objective function needs to be constructed, and the specific content is as follows:
wherein (x) i ,y i ) Is the coordinate of beacon i.
S72, converting the problem of the formula (9) into a quadratic function, namely:
wherein Q is a symmetric positive definite matrix, T represents transposition, b is a corresponding constant matrix, and x is a coordinate vector;
s73, initialization, and the estimated coordinates (x) of the unknown node e e ,y e ) Set to an initial value, x m Represents the m-th iteration coordinate solution vector, d m Is the direction vector of the m-th iteration, r m Is the residual vector of the mth time;
calculating the direction vector g of the gradient 1 ：
g 1 ＝Qx 1 -b (11)
Let the negative of the gradient direction be the search direction d 1
d 1 ＝r 1 ＝-g 1 (12)
S74, an iteration process: d is continuously adjusted in the iterative process 1 ,d 2 ,d 3 ...d m Constructing a conjugate direction of a Black-plug matrix (Hessian matrix) as a search direction by a Gram-Schmidt (Gram-Schmidt) method; wherein the step length is:
position of the next point:
x m+1 ＝x m +α m d m (14)
updating the gradient:
r m+1 ＝r m -x m Qd m (15)
s75, if k = n, obtaining the current optimal solution x m+1 If k is not equal to n, continue to r m+1 Projection to obtain beta m ：
And determining the next search direction d m+1 ：
d m+1 ＝r m+1 -β m d m (17)
Continuing to execute S74;
s76, the process is repeated in such a way until x is finally solved m Terminating the iterative process by the optimal solution; and obtaining the final coordinate of the unknown node.
Compared with the prior art, the improved wireless sensor network DV-HOP positioning optimization method based on M-CG provided by the invention has the beneficial effects of the following patent law meanings:
1) The invention introduces the Manhattan distance on the basis of the DV-HOP algorithm, adopts the combination of the Manhattan distance and the Euclidean distance as a reference to reflect the curve path propagation distance of the HOP count of the original beacon nodes more truly, and calculates a new HOP count corresponding to the straight line distance between the beacon nodes by taking the Manhattan distance and the Euclidean distance as the reference, wherein the new HOP count corrects the HOP count error problem of the DV-HOP model to a certain extent, and corrects the average HOP distance calculated in the next step, thereby improving the positioning accuracy.
2) Aiming at the problem that the estimated coordinates of unknown nodes calculated by the existing DV-HOP algorithm by using a maximum likelihood estimation method are large in error, the CG algorithm is introduced on the basis of the obtained estimated coordinates, the coordinates of the unknown nodes are subjected to fast iterative refinement again, conjugation and the steepest descent method are combined, a group of conjugation directions are constructed by using the gradient of the known point, the optimal solution of searching pixels is carried out along the group of conjugation directions, and the positioning accuracy is improved.
3) Compared with other algorithms, the method has the advantages of small storage, low time complexity, higher convergence, secondary termination and the like. In addition, the dependence on hardware is low without ranging, the cost is low, and the positioning precision is improved.
Drawings
Fig. 1 is a flowchart of a DV-HOP location optimization method for a wireless sensor network based on M-CG improvement provided by the present invention.
Fig. 2 is an explanatory effect diagram of a HOP path, a manhattan path and a euclidean path generated by a positioning method in the M-CG-based improved wireless sensor network DV-HOP positioning optimization method provided by the invention.
Fig. 3 is a flowchart of a CG algorithm in the method for optimizing DV-HOP positioning of a wireless sensor network based on M-CG improvement according to the present invention.
Detailed Description
The following describes a wireless sensor network DV-HOP location optimization method based on M-CG improvement in more detail by specific embodiments:
example 1
Referring to fig. 1-3, the method for optimizing DV-HOP location of a wireless sensor network based on M-CG improvement of the present embodiment includes:
s1, initializing communication radiuses of nodes in a network, finishing random network topology deployment of beacon nodes and unknown nodes, and broadcasting self information and hop values of all beacon nodes to surrounding neighbor nodes;
step S1 includes:
s11, initializing the communication radius of the nodes in the network, and finishing network topology deployment of the beacon nodes and unknown nodes;
s12, broadcasting a data packet containing self number, position information and hop value by a beacon node in the wireless sensor network in a flooding mode, wherein the initial value of each beacon node hop value is set to be zero;
s2, an unknown node in the network receives the information packet broadcast by the neighbor node and forwards the information packet to enable all nodes in the network to obtain the minimum hop count from the reference beacon node;
step S2 includes:
s21, after receiving the information packet of the beacon node, the neighbor node of the beacon node adds one to the hop count value in the information packet and forwards the hop count value to the neighbor node of the next hop;
s22, the node receiving the information packet selects and stores the information packet with the minimum hop value from a plurality of information packets with the same beacon node number received by the node; the node adds one to the hop count value in the selected information packet and forwards the hop count value to the neighbor node of the next hop; repeating the step S22 until flooding is finished;
s3, calculating a new minimum hop count between each beacon node by adopting the mean value of the Manhattan distance and the Euclidean distance; calculating a new minimum hop count from the unknown node to the beacon node;
step S3 includes:
s31, according to the step S1, because each node in the network is randomly distributed, the hop number of the node between two points is selected randomly, the route between the two points is not a straight line, and the introduced Manhattan distance | x between the beacon nodes is adopted i -x j |+|y i -y j I and Euclidean distance
The average value of (a) is used as a curve path distance which truly reflects the original minimum hop count, and as shown in fig. 2, a new minimum hop count value of a straight line path between two nodes is correspondingly calculated, which specifically comprises:
wherein, the first and the second end of the pipe are connected with each other,
representing the original minimum number of hops between beacon i and beacon j,
the new minimum hop count between the beacon node i and the beacon node j is represented by i, j belongs to B, i is not equal to j, and B is the number of unknown nodes; (x) i ,y i ) And (x) j ,y j ) Coordinates of a beacon node i and a beacon node j are respectively represented;
calculating the new minimum hop count between beacon i and beacon j
Comprises the following steps:
s32, correcting the original hop count between the unknown node e and the beacon node i to obtain a new minimum hop count between the unknown node e and the beacon node i, which specifically comprises the following steps:
wherein e is an unknown node to be positioned,
for the original minimum number of hops between unknown node e and beacon i,
is the new minimum number of hops between the unknown node e and the beacon node i.
S4, calculating the average hop distance, and optimizing the average hop distance by using the normalization of the new minimum hop value among all nodes as a weight;
step S4 includes:
s41, calculating the average hop distance of the beacon nodes according to the new minimum hop count obtained among the beacon nodes, specifically:
wherein, hopsize i Representing the average per-hop distance of the beacon i,
representing the new minimum number of hops between beacon i and beacon j.
S42, in order to enable the average hop distance of the positioning node to be closer to the actual average hop distance, normalizing the minimum hop value obtained in S3 through each beacon node, wherein the specific formula is as follows:
wherein, B is the number of the beacon nodes.
S43, optimizing the delta serving as a weight to obtain a new average hop distance, which specifically comprises the following steps:
s5, calculating the distance between each unknown node and the beacon node according to the product of the minimum hop count of the unknown node obtained in the S3 and the average hop distance obtained in the S4;
in step S5, a new minimum hop count of the unknown node e and the beacon node i may be obtained in S32, and the distance between each unknown node and the beacon node may be obtained by calculating the product of the minimum hop count and the average per-hop distance obtained in S43;
s6, calculating the estimated coordinates of the unknown nodes by using a maximum likelihood estimation method;
in step S6, let the coordinates of unknown node e in the network be (x) e ,y e ) (ii) a Let the coordinate of beacon i be (x) i ,y i ) (ii) a Estimating the distance d between the unknown node e and the beacon node i ei Calculating the estimated coordinates of the unknown nodes by using a following equation set, wherein the specific equation set comprises:
s7, as shown in FIG. 3, iterative refinement is carried out on the estimated coordinates of the unknown nodes by using a CG algorithm:
step S7 includes:
s71, because an error zeta exists between the distance between the unknown node and the beacon node, in order to minimize the error, an objective function needs to be constructed, and the specific content is as follows:
wherein (x) i ,y i ) Is the coordinate of beacon i.
S72, converting the problem of the formula (9) into a quadratic function, namely:
wherein Q is a symmetric positive definite matrix, T represents transposition, b is a corresponding constant matrix, and x is a coordinate vector;
s73, initialization, and the estimated coordinates (x) of the unknown node e e ,y e ) Set to an initial value, x m Represents the m-th iteration coordinate solution vector, d m Is the direction vector of the m-th iteration, r m Is the residual vector of the mth time.
Calculating the direction vector g of the gradient 1 ：
g 1 ＝Qx 1 -b (11)
Let the negative of the gradient direction be the search direction d 1 ：
d 1 ＝r 1 ＝-g 1 (12)
And S74, an iterative process. D is continuously adjusted in the iterative process 1 ,d 2 ,d 3 ...d m The conjugate direction of a Black-plug matrix (Hessian matrix) is constructed by a Gram-Schmidt (Gram-Schmidt) method and is used as a search direction.
Wherein the step length is:
position of the next point:
x m+1 ＝x m +α m d m (14)
updating the gradient:
r m+1 ＝r m -x m Qd m (15)
s75, if k = n, obtaining the current optimal solution x m+1 If k is not equal to n, continue to r m+1 Projection to obtain beta m ：
And determining the next search direction d m+1 ：
d m+1 ＝r m+1 -β m d m (17)
Continuing to execute S74;
s76, the process is repeated in such a way until x is finally solved m The optimal solution is terminated by the iterative process, and the final unknown is obtainedCoordinates of the nodes.
And calculating a new hop number corresponding to the straight-line distance between the beacon nodes by taking the Manhattan distance and the Euclidean distance in combination with the original hop number path distance between two nodes as reference, and iteratively refining the estimated coordinates of the unknown nodes by using a CG algorithm, thereby improving the robustness and the positioning accuracy of the algorithm.
It should be understood that these examples are for illustrative purposes only and are not intended to limit the scope of the present invention. Further, it should also be understood that various alterations, modifications and/or variations can be made to the present invention by those skilled in the art after reading the technical content of the present invention, and all such equivalents fall within the protective scope defined by the claims of the present application.
It will be appreciated by those skilled in the art that the invention may be embodied in other specific forms without departing from the spirit or essential characteristics thereof. The embodiments disclosed above are therefore to be considered in all respects as illustrative and not restrictive. All changes which come within the scope of or equivalence to the invention are intended to be embraced therein.
Claims (8)
1. The improved DV-HOP positioning optimization method of the wireless sensor network based on the M-CG is characterized by comprising the following steps:
s1, initializing communication radiuses of nodes in a network, finishing random network topology deployment of beacon nodes and unknown nodes, and broadcasting self information and hop values of all beacon nodes to surrounding neighbor nodes;
s2, an unknown node in the network receives the information packet broadcast by the neighbor node and forwards the information packet to enable all nodes in the network to obtain the original minimum hop count from the reference beacon node;
s3, calculating a new minimum hop count between each beacon node by adopting the mean value of the Manhattan distance and the Euclidean distance; calculating the new minimum hop count from the unknown node to the beacon node;
s4, calculating the average hop distance, and optimizing the average hop distance by using the normalization of the new minimum hop value among all nodes as a weight;
s5, calculating the distance between each unknown node and the beacon node according to the product of the new minimum hop count between the unknown nodes obtained in the S3 and the average hop distance obtained in the S4;
s6, calculating the estimated coordinates of the unknown nodes by using a maximum likelihood estimation method;
and S7, carrying out iterative refinement on the estimated coordinates of the unknown nodes by using a CG algorithm.
2. The M-CG-based improved wireless sensor network DV-HOP location optimization method according to claim 1, wherein: the step S1 includes:
s11, initializing the communication radius of the nodes in the network, and finishing network topology deployment of the beacon nodes and unknown nodes;
s12, broadcasting a data packet containing self number, position information and hop value by the beacon nodes in the wireless sensor network in a flooding mode, wherein the initial value of each beacon node hop value is set to be zero.
3. The M-CG-based improved wireless sensor network DV-HOP location optimization method according to claim 1, wherein: the step S2 includes:
s21, after receiving the information packet of the beacon node, the neighbor node of the beacon node adds one to the hop count value in the information packet and forwards the hop count value to the neighbor node of the next hop;
s22, the node receiving the data packet selects and stores the information packet with the minimum hop value from the plurality of information packets with the same beacon node number received by the node; the node adds one to the hop count value in the selected information packet and forwards the hop count value to the neighbor node of the next hop; step S22 is repeated until flooding is completed.
4. The M-CG-based improved wireless sensor network DV-HOP location optimization method according to claim 1, wherein: the step S3 includes:
s31, according toIn the step S1, because each node in the network is randomly distributed, the hop number of the node between two points is selected randomly, the route between the two points is not a straight line, and the introduced Manhattan distance | x between the beacon nodes is adopted i -x j |+|y i -y j I and Euclidean distance
If the mean value of the two nodes is taken as the curve path distance which truly reflects the original minimum hop count, a new minimum hop count value of a straight line path between the two nodes is correspondingly calculated, and the method specifically comprises the following steps:
wherein the content of the first and second substances,
representing the original minimum number of hops between beacon i and beacon j,
the new minimum hop count between the beacon node i and the beacon node j is represented by i, j belongs to B, i is not equal to j, and B is the number of unknown nodes; (x) i ,y i ) And (x) j ,y j ) Coordinates of a beacon node i and a beacon node j are respectively represented;
calculating the new minimum hop count between the beacon node i and the beacon node j
Comprises the following steps:
s32, correcting the original hop count between the unknown node e and the beacon node i to obtain a new minimum hop count between the unknown node e and the beacon node i, which specifically comprises the following steps:
wherein e is an unknown node to be positioned,
for the original minimum number of hops between unknown node e and beacon i,
is the new minimum number of hops between the unknown node e and the beacon node i.
5. The M-CG-based improved wireless sensor network DV-HOP location optimization method according to claim 4, wherein: the step S4 includes:
s41, calculating the average hop distance of the beacon nodes according to the new minimum hop count obtained among the beacon nodes, specifically:
wherein, hopsize i Representing the average per-hop distance of the beacon i,
representing the new minimum number of hops between beacon i and beacon j;
s42, in order to enable the average hop distance of the positioning node to be closer to the actual average hop distance, normalizing the new minimum hop value obtained in S3 by each beacon node, wherein the specific formula is as follows:
b is the number of the beacon nodes;
s43, optimizing the delta serving as a weight to obtain a new average hop distance, which specifically comprises the following steps:
6. the M-CG-based improved wireless sensor network DV-HOP location optimization method according to claim 5, wherein: in step S5, the distance between each unknown node and the beacon node is calculated;
s51, a new minimum hop count of the unknown node e and the beacon node i can be obtained through S32, and the product of the minimum hop count and the average hop distance obtained through S43 is calculated to obtain the distance between each unknown node and the beacon node;
7. the M-CG-based improved wireless sensor network DV-HOP location optimization method according to claim 1, wherein: in step S6, the coordinates of the unknown node are calculated by the maximum likelihood estimation method:
s61, calculating an estimated coordinate of the unknown node by using a maximum likelihood estimation method; let the coordinate of unknown node e in the network be (x) e ,y e ) (ii) a Let the coordinate of beacon i be (x) i ,y i ) (ii) a Estimating the distance d between the unknown node e and the beacon node i ei Calculating the estimated coordinates of the unknown nodes by using a following equation set, wherein the specific equation set comprises:
8. the M-CG-based improved wireless sensor network DV-HOP location optimization method according to claim 1, wherein: the step S7 includes:
s71, because an error zeta exists between the distance between the unknown node and the beacon node, in order to minimize the error, an objective function needs to be constructed, and the specific content is as follows:
wherein (x) i ,y i ) Is the coordinate of the beacon node i;
s72, converting the problem of the formula (9) into a quadratic function, namely:
wherein Q is a symmetric positive definite matrix, T represents transposition, b is a corresponding constant matrix, and x is a coordinate vector;
s73, initializing, and estimating the coordinate (x) of the unknown node e e ,y e ) Set to an initial value, x m Representing the m-th iteration coordinate solution vector, d m Is the direction vector of the m-th iteration, r m Is the residual vector of the mth time;
calculating the direction vector g of the gradient 1 ：
g 1 ＝Qx 1 -b (11)
Let the negative of the gradient direction be the search direction d 1
d 1 ＝r 1 ＝-g 1 (12)
S74, an iteration process: d is continuously adjusted in the iterative process 1 ,d 2 ,d 3 ...d m Constructing a conjugate direction of a Black-plug matrix (Hessian matrix) as a search direction by a Gram-Schmidt (Gram-Schmidt) method; wherein the step length is:
position of the next point:
x m+1 ＝x m +α m d m (14)
updating the gradient:
r m+1 ＝r m -x m Qd m (15)
s75, if k = n, obtaining the current optimal solution x m+1 If k is not equal to n, continue to r m+1 Projection to obtain beta m ：
And determining the next search direction d m+1 ：
d m+1 ＝r m+1 -β m d m (17)
Continuing to execute S74;
s76, the process is repeated in such a way until x is finally solved m Terminating the iterative process; and obtaining the final coordinate of the unknown node.
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