CN101834717B - Parallel computing method capable of expanding precision Logistic chaotic sequence - Google Patents

Parallel computing method capable of expanding precision Logistic chaotic sequence Download PDF

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CN101834717B
CN101834717B CN 201010152456 CN201010152456A CN101834717B CN 101834717 B CN101834717 B CN 101834717B CN 201010152456 CN201010152456 CN 201010152456 CN 201010152456 A CN201010152456 A CN 201010152456A CN 101834717 B CN101834717 B CN 101834717B
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CN101834717A (en
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刘嘉辉
宋大华
陈德运
乔佩利
王卫兵
李岩
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Harbin University of Science and Technology
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Abstract

The invention discloses a parallel computing method capable of expanding a precision Logistic chaotic sequence, comprising the following steps: (1) saving initial parameters x0 and mu integer quantization to an one-dimensional array, and arranging the precision required to be achieved or designating iterative times; (2) the first part of the parallel computational algorithm x=x*(1-x): utilizing a matrix to save and compute the intermediate result, modifying the length of a dynamic array and saving the computation result to the dynamic array; (3) the second part of the parallel computational algorithm x=mu*x: saving the computation result to the dynamic array; (4) and if the precision or the iterative times achieves the required value, acquiring the chaotic sequence in the dynamic array, thus the computation is finished, and otherwise, transferring to step (2). The method of the invention fully utilizes the properties of a chaotic system, provides the chaotic sequence of a more expansive mapping space, can be used in the fields of secret communication, information safety and so on, and is especially suitable for encryption, storage and network transmission of graph, image and multimedia and other information.

Description

But the parallel calculating method of extended precision Logistic chaos sequence
Technical field
The invention belongs to data security, network security technology field, but specifically utilize parallel computing to realize obtaining based on extended precision the method for chaos random number series.
Background technology
After at first having proposed the concept of chaos in the research in atmospheric science in 1963, chaos has all obtained application in various degree in every field since Edward Lorenz.Chaotic motion refers to be confined to the highly unstable motion of the confined space in deterministic system; Chaos is produced by the certainty equation, as long as equation parameter and initial value are determined just can reappear chaos phenomenon.
The maximum characteristics of chaos system are that the evolution of system is very responsive to initial condition, are uncertain from the future behaviour of Long Significance system.Shannon, U.S. mathematician Shannon points out: if can produce in some way a random sequence, this sequence determined by key, and a minor variations of any input value all has considerable influence to output, utilizes such sequence just can be encrypted.Chaos system exactly meets this requirement.Chaos system has sensitivity to initial, i.e. any small variation can make chaos sequence afterwards produce unpredictable variation; Therefore, the essence of chaos encryption is to encrypt with the sequence that is difficult to predict rather than with the number that is difficult to guess.
In 1989, at first British mathematician Matthews proposed to use discrete chaos dynamic in cryptography chaos at the research origin of field of cryptography, and chaos encryption is more and more paid close attention to by people afterwards, and had obtained development rapidly.1996, the people such as Feldmann proposed the universal model of safe chaotic communication.1997, Zhou etc. proposed the chaotic model of the Piecewise linear chaotic map (Piecewise Linear Chaotic Map) under limited precision.1998, Fridrich proposed the chaotic model of 2 D chaotic mapping.Afterwards, the models such as multidimensional, mixing have appearred again in the research of chaotic model.
1976, Robert May used the demographic model in Logistic equation (Logistic equation) the research ecosystem; Also someone is cited as Logistic mapping (Logistic map), and it is used in ecology widely, in the subjects such as economics.The chaotic model of Logistic mapping produces simple with it, fast, the characteristics such as realization that are conducive to computer are commonly used in the information encryption field.
Chaotic model is widely used in multimedia messagess such as encrypting graph image, voice, video.At present, the most of algorithm that utilizes chaotic maps to carry out image encryption is: first select a kind of chaotic model, utilize chaotic maps to generate chaos sequence, then chaos sequence is made quantification treatment, utilize new sequence and the original image that produces necessarily to process.Therefore, the good randomness of chaos sequence has guaranteed the safety of information.
The Piecewise linear chaotic map that Li in 2003 etc. have verified Zhou is not safe enough from the angle of strict chaos, and point out that the dynamics of chaos system is with slump of disastrous proportions under the condition of limited computational accuracy (Finite Computing Precision).Deteriorate to the chaos of periodic sequence, lost the distinctive stochastic behaviour of chaos system; Therefore, can not guarantee the safety of enciphered data.
In order to overcome the degenerate problem of chaos sequence under limited precision, adopt the method for scramble, disturbance and multidimensional chaotic maps can reduce the speed that chaos sequence is degenerated, increased sequence period length under given conditions; But, can not tackle the problem at its root.
But solved the restriction of the precision of chaos sequence based on the chaotic model of extended precision, satisfied different user to the requirement of precision, both different encryption requests.Particularly for requiring fast, the transmission of real-time network multimedia data both can guarantee the speed transmitted, also can guarantee the safety of enciphered data.
Yet, but what need to face based on the chaotic model of extended precision is the large-scale calculations problem, and the best solution of large-scale calculations problem is to adopt parallel computing.
The fund that the development need of parallel computing is powerful and professional and technical personnel, reason is the price of the supercomputer of parallel computation or a group of planes far above common computer, and the programming model under parallel environment just can be write by the programmer of professional training also only.These effects limit parallel computing universal.Yet the development that appears as parallel computation of polycaryon processor has brought favourable turn.
Many kernels refer to integrated two or more complete computing engines (kernel) in one piece of processor.Since two thousand five, along with the appearance of dual core processor; Four core processors were released in 2006, and multi-core technology was commonly used in PC in recent years, and the ability of PC deal with data is greatly improved.The processor development is forwarded to gradually by pure frequency upgrading on the direction of multinuclear computing and executed in parallel.The use of polycaryon processor has improved the ability of computer processing data, therefore, takes full advantage of the characteristic of multi-core computer, and the software product that exploitation has a concurrency is software development direction from now on.
The parallel encryption technology is in order to adapt to the trend of present parallelization, to the improvement of traditional encryption technology.Exploitation parallel encryption algorithm has certain difficulty, and its Parallel Implementation needs the programmer to be familiar with the environment of parallel system and the programmed method of parallel Programming, and the difficulty of multiple programming is much larger than traditional sequential program design.Be fit to do the programming based on sharing storage (thread) on multinuclear, SMP computing node, and be more suitable for adopting the programming of message transmission between node.As using the mixed develop approach of MPI and OpenMP, this mixed develop approach can make already very complicated design of Parallel Algorithms task become more difficult, and requires the designer of concurrent program all very familiar to these two kinds of programming models.This has just greatly increased the difficulty of parallel Programming, has improved the requirement to parallel Programming person.How the advantage with the two combines well, and avoids the difficulty of the parallel program development that mixed develop approach brings as far as possible.How bringing into play better the performance advantage of processor under multi-core environment, the demand that satisfies the application of the data encryption under parallel environment has just become the current subject matter that faces.At present, China is also in the research of constantly strengthening the aspect such as parallel data encryption, but the research in this field and industrialization are in starting stage of development upwards.
Summary of the invention
The object of the invention is to utilize the good pseudo-random characteristics of chaos system, but under the prerequisite of extended precision, use parallel computing to obtain desirable chaos pseudo random ordered series of numbers.
But the invention provides a kind of parallel calculating method of extended precision Logistic chaos sequence, its step is as follows:
1, the initial parameter x of Logistic equation 0, the μ integer quantisation is saved in one-dimension array, sets to require the precision that reaches or the number of times of specifying iteration;
2, parallel algorithms first: x=x * (1-x), utilize matrix to preserve and calculate intermediate object program revises the length of dynamic array X, preserves result of calculation in dynamic array X;
3, parallel algorithms second portion: x=μ * x, utilize matrix to preserve and calculate intermediate object program, revise the length of dynamic array X, preserve result of calculation in dynamic array X;
If 4 precision or the iterations value of meeting the requirements of dynamically are in array X and obtain chaos sequence, calculate and finish; Otherwise, forward step 2 to.
In above-mentioned steps 1, establish initial parameter x 0Value be x, x can be expressed as
x = Σ i = 1 n s i × D - i , D = 10 , s i ∈ { 0,1 , · · · , 9 } , s n ≠ 0 , n = 1,2 , · · ·
N is the number of significant digit after parameter value x decimal point, i.e. precision.
If the value of initial parameter μ is designated as
μ = Σ i = 0 n u i × D - i , D = 10 , u i ∈ { 0,1 , · · · , 9 } , u n ≠ 0 , n = 0,1 , · · ·
The vectorial X={s of row 1, s 2..., s nBe kept in dynamic array X, dynamically first element of array X is preserved s 1, i.e. X[0]=s 1, X[1]=s 2... by that analogy.The element sum of the vectorial X of row, namely dynamically the length of array X is designated as L x
The vectorial U={u of row 1, u 2..., u nBe kept in array U, first element of array U is preserved u 1, i.e. U[0]=u 1, U[1]=u 2... by that analogy.
In above-mentioned steps 2, establish y=1-x n=1-x has
y = Σ i = 1 n r i × D - i , r i ∈ { 0,1 , · · · , 9 } , n = 1,2 , · · ·
Figure GSA00000094128500032
The vectorial Y={r of row 1, r 2..., r nBe stored in dynamic array Y, dynamically first element of array Y is preserved r 1, i.e. Y[0]=r 1, Y[1]=r 2... by that analogy.
If row vector M i=r N-i+1* X, 1≤i≤n, and Mi has following form:
Mi＝{mi 1，mi 2，...mi n}.
Matrix M is preserved the results of intermediate calculations of x * (1-x), has following form:
M = 0 0 · · · · · · 0 0 s 1 r n s 2 r n · · · · · · s n - 1 r n s n r n 0 0 · · · · · · 0 s 1 r n - 1 s 2 r n - 1 · · · · · · s n - 1 r n - 1 s n r n - 1 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 s 1 r 2 s 2 r 2 · · · · · · s n - 1 r 2 s n r 2 0 · · · · · · 0 0 s 1 r 1 s 2 r 1 · · · · · · s n - 1 r 1 s n r 1 0 0 · · · · · · 0 .
In every row of matrix M, other elements are zero except the element of row vector M i.Note M is:
M = m 11 m 12 · · · m 1 n m 1 ( n + 1 ) · · · m 1 ( 2 n ) m 21 m 22 · · · m 2 n m 2 ( n + 1 ) · · · m 2 ( 2 n ) . . . . . . . . . . . . . . . . . . . . . m n 1 m n 2 · · · m nn m n ( n + 1 ) · · · m n ( 2 n ) n × 2 n .
The initialization matrix M is null matrix: m ik=0,1≤i≤n, 1≤k≤2n.I is in matrix M for the storage line vector M, namely preserves the results of intermediate calculations of x * (1-x):
m ik＝mi j，k＝j+n-i+1，1≤i，j≤n.
Calculate this result:
α i＝(η i+d i)(mod D)，1≤i≤2n；
η i = Σ j = 1 n m ji ;
d i-1＝int[(η i+d i)/D]，1≤i≤2n；
Function int[] for getting the integer part of result of calculation, establish d 2n=0.
Row vector M i can be used as independently parallel computation in the different processors of calculating section under parallel environment.
Supposing has N in parallel system ProsIndividually can use i the corresponding numbering of processor p from processor i, as follows based on the parallel algorithm of Message-Passing Model:
(1) work as N Pros〉=L xThe time, master processor transmits the vectorial X of row and r iArrive from processor p iThe reception buffer memory in, vector M i is at processor p at once iMiddle calculating; After completing from processor calculating, transmission result of calculation is to master processor.
(2) work as N Pros＜L xThe time, in parallel system, because the sum of row vector M i surpasses from the processor sum, therefore, some processor need to calculate the capable vector M i more than 2 or 2.The numbering of row vector M i and processor has following relation:
p i＝(i-1)(mod N pros)+1(i＝1，2，...；p i＝1，2，...，N pros).
I is the numbering of row vector M i.Transmit result of calculation to master processor after capable vector M i is completed in processor calculating, complete until all capable vector M i calculate.
The result of calculation of x * (1-x) is preserved in the vector T of being expert at, and form is as follows:
T＝{α 1，α 2，...α n，α n+1，...α 2n}.
The row vector T is kept in dynamic array Temp, and dynamically the length of array Temp is 2n.
In order to carry out the calculating of next part, the length that enlarges dynamic array X is 2n, copies data element in dynamic array Temp in dynamic array X, release dynamics array Temp.
In above-mentioned steps 3, the process of parallel computation x=μ * x is except partial parameters, matrix and array length difference, and computational methods are identical with algorithm first.
In above-mentioned steps 4, setting parameter x nPrecision be L n, the precision of initial parameter μ is L u, iterations and precision have following relation:
L n+1＝2×L n+L u；L n，L u≠0；n＝0，1，...
When this iteration obtains precision that precision sets greater than the user, can take the method for truncation to process, namely cast out remaining number of significant digit.
Characteristics of the present invention:
1, utilize dynamic array to preserve chaos equation result of calculation, overcome the short period phenomenon of chaos sequence under limited precision, but the chaos equation based on extended precision has increased mapping space, makes the chaos sequence of generation more close to perfect condition, can take full advantage of the characteristic of chaos system.
2, the present invention is based on parallel computing, take full advantage of the present advantage of universal polycaryon processor gradually, realize producing fast and efficiently chaos sequence, effective solution can be provided for the data security under parallel environment.
Description of drawings
Fig. 1 is the algorithm pattern of Logistic equation;
Fig. 2 a is that precision equals 1 o'clock iteration oscillogram;
Fig. 2 b is that precision equals 2 o'clock iteration oscillograms;
Fig. 2 c is that precision equals 4 o'clock iteration oscillograms;
Fig. 3 a is that precision equals 200 oscillograms of 6 o'clock iteration;
Fig. 3 b is that precision is 200, error is the significance bit oscillogram of 0.000001 o'clock;
Fig. 3 c is that precision is 50,1000 and the probability statistics figure of 10000 o'clock;
Fig. 4 is based on the block diagram of message passing interface (MPI) and modularized design;
Fig. 5 a is the former figure of Lena;
Fig. 5 b is the pixel scatter chart of the former figure of Lena;
Fig. 5 c is encrypted image;
Fig. 5 d is the pixel scatter chart of encrypted image.
Embodiment
The present invention is described in further detail below in conjunction with drawings and Examples, but embodiments of the present invention are not limited to this.
The characteristic of chaos system has: chaos system is to initial condition extreme sensitiveness; The minor variations of initial value has a great impact the value tool of subsequently generation.Chaos system has self-similarity.The chaos form that the part is chosen is fully similar to integral body, and in theory, chaos sequence is chaos on whole space; Therefore, the subsequence of choosing at certain local space is also chaos.The randomness of determining.The dynamic behavior of chaos system is extremely complicated, does not meet the probability statistics principle; Therefore, be difficult to reconstruct and prediction.
When 3.5699...＜μ≤4, the Logistic equation enters chaos state; But due to the limited precision problem that calculates, the sequence that often obtains has become the sequence in cycle.Suppose x 0=0.1, μ=3.9; Given accuracy be respectively 1,2 and the iteration oscillogram of 4 o'clock see Fig. 2 a, Fig. 2 b and Fig. 2 c.In the situation that limited precision, the Logistic mapping cycle occurred through after certain iterations.Although chaos has good pseudo-randomness,, the random number series that produces under limited precision deteriorates to periodic sequence.
Suppose x 0=0.1, μ=3.9; The maximal accuracy of Logistic equation and the relation of iterations see Table 1.
Table 1 Logistic Iterates of maps number of times and maximal accuracy relation
Iterations Maximal accuracy Iterations Maximal accuracy
1 3 11 4095
2 7 12 8191
3 15 13 16383
4 31 14 32767
5 63 15 65535
6 127 16 131071
7 255 17 262143
8 511 18 524287
9 1023 19 1048575
10 2047 20 2097151
Through an iteration, maximal accuracy is 3; Through 10 iteration, maximal accuracy is 2047.In theory, chaos sequence is no periodic on whole space, and is uncertain.Obviously, along with the raising of precision, mapping space enlarges, and the complexity of calculating increases.Adopt traditional computational methods can't satisfy the large-scale calculations task, therefore, the algorithm of design parallel computation Logistic equation can solve the problem that enlarges mapping space, also can raise the efficiency.Meanwhile, but based on the Logistic equation of extended precision, can produce desirable chaos random number series.
The auto-correlation of chaotic motion is approximately the δ function, and the chaos sequence cross-correlation that is produced by two different initial values is 0, and its characteristic is close to white noise.In Fig. 3 a, the precision of Logistic equation equals 6, μ=3.98765.Work as x 0=0.1 o'clock,
A＝{α 1，α 2，...α n}，
α iRepresent the value of the i time iteration.Work as x 0=0.100001 o'clock, B={ β 1, β 2... β n.Through 200 iteration, can find out, differed 0.000001 o'clock at two groups of initial values, chaos system is to initial condition extreme sensitiveness; But in initial iteration several times, the difference of two class values is near 0.000003, and therefore, initial iterative value can not be used as random number series.In Fig. 3 b, precision is 200, through 100 iteration; Work as x 0, obtain chaos sequence ξ at=0.1 o'clock 1Work as x 0, obtain chaos sequence ξ at=0.100001 o'clock 2, this shows, chaos sequence has good pseudo-random characteristics.
S in the chaos sequence that produces i∈ 0,1 ... and 9}.0,1 ..., 9 probability that occur are designated as P (0), P (1) ..., P (9), in theory, P (0)=P (1)=...=P (9)=1/D.Suppose μ=3.98765, x 0=0.12345, precision is respectively 50 (S1), 1000 (S2) and 10000 (S3), and through 50 iteration, 0,1 ..., 9 probability distribution graph that occur are seen Fig. 3 c.Can find out, along with the increase of precision, i.e. the expansion of mapping space will obtain desirable chaos random number series.
Example 1: based on the realization of message passing interface (MPI) and modularized design.
Based on the step of the realization of message passing interface (MPI) and modularized design as shown in Figure 4:
1, the initial parameter x of Logistic equation 0, the μ integer quantisation is saved in one-dimension array, sets to require the precision that reaches or the number of times of specifying iteration; Parallel system obtains the current information from processor that can use, perhaps specifies in parallel system the number from processor.After using MPI_Init () function to carry out the MPI initialization, can use MPI_Comm_rank () function to obtain the numbering of processor in parallel system, i.e. p iCan use processor number in MPI_Comm_size () function acquisition system.
2, parallel algorithms first: x=x * (1-x), utilize matrix to preserve and calculate intermediate object program revises the length of dynamic array X, preserves result of calculation in dynamic array X.
The parallel computation function is based on the parallel algorithms first of modularized design and the main function of second portion, and it has two parameters: the numbering from processor of the calculating section of algorithm and execution parallel computation.
Work as N Pros〉=L xThe time, master processor transmits the vectorial X of row and r iArrive from processor p iThe reception buffer memory in, vector M i is at processor p at once iMiddle calculating; After completing from processor calculating, transmission result of calculation is to master processor.
Work as N Pros＜L xThe time, in parallel system, because the sum of row vector M i surpasses from the processor sum, therefore, some processor need to calculate the capable vector M i more than 2 or 2.The numbering of row vector M i and processor has following relation:
p i＝(i-1)(mod N pros)+1(i＝1，2，...；p i＝1，2，...，N pros).
I is the numbering of row vector M i.
Transmit result of calculation to master processor after capable vector M i is completed in processor calculating, complete until all capable vector M i calculate.Master processor waits for that all capable vector M i calculate and completes, and namely Process Synchronization, can use function MPI_Barrier () to realize.Then, master processor calculates the result of x * (1-x) and preserves in the vector T of being expert at, and the row vector T is kept in dynamic array Temp, and dynamically the length of array Temp is 2L xIn order to carry out the calculating of next part, the length that enlarges dynamic array X is 2L x, copy data element in dynamic array Temp in dynamic array X.
3, parallel algorithms second portion: x=μ * x, utilize matrix to preserve and calculate intermediate object program, revise the length of dynamic array X, preserve result of calculation in dynamic array X.
What the parallel computation function calculated at this moment is the second portion of algorithm, and computational methods are identical with first part of calculating.Master processor uses function MPI_Barrier () to wait for that all capable vector M i calculate and completes, and then, master processor calculates the result of μ * x and preserves in the vector T of being expert at, and the row vector T is kept in dynamic array Temp, and the length of dynamic array Temp is L x+ L uThe length that enlarges dynamic array X is L x+ L u, copy data element in dynamic array Temp in dynamic array X.
If 4 precision or the iterations value of meeting the requirements of dynamically are in array X and obtain chaos sequence, calculate and finish; Otherwise, forward step 2 to.
Use function MPI_Send () and MPI_Recv () to receive and send message between multiprocessor based on the realization of message passing interface (MPI) and modularized design, the data of transmission are as the message transmission.
Example 2: realize the encryption to image.
But the basic thought based on the algorithm of the chaos random number series encrypted image of extended precision is to segment the image into several parts (data slice), the cipher key number that the chaos random number series that the length of tentation data sheet equals to generate can generate, data slice is carried out byte encrypt, concrete step is as follows:
1, the initial parameter x of Logistic equation 0, μ, the number of data fragmentation etc. can be used as the cryptopart of user's input.Initial parameter x 0, the μ integer quantisation is saved in one-dimension array.
2, parallel algorithms first: x=x * (1-x);
3, parallel algorithms second portion: x=μ * x;
If 4 precision reach the generation cipher key number, forward step 5 to; Otherwise, forward step 2 to;
5, the chaos sequence that obtains need to be combined into key, and the way of combination can be following method: adjacent n element s 1, s 2..., s nAs a key K
K = ( Σ i = 1 n s i × D i - 1 mod 256 ) , D = 10 , s i ∈ { 0,1 , · · · , 9 } , n = 1,2 , · · ·
6, with the secret key encryption current data sheet that generates;
If 7 all data slice are encrypted complete, this algorithm finishes; Otherwise, forward step 2 to.
What show in Fig. 5 a and Fig. 5 b is the former figure of lena and its pixel scatter chart, obtains Fig. 5 c through encryption.What Fig. 5 d showed is the pixel scatter chart of encrypted image.The key space that the method generates is large, make the safety of image after encryption higher, and cipher round results is even, is difficult for being cracked.
Example 3: network encryption traffic model.
Chaos encryption algorithm is symmetric encipherment algorithm, namely uses same algorithm to be encrypted/to decipher at transmitting terminal and receiving terminal.
At first transmitting terminal and receiving terminal determine the initial parameter x of Logistic equation 0, μ, and precision or iterations, they can be used as the cryptopart of user's input.Transmitting terminal can be encrypted processing to the information that needs transmit, then by Internet Transmission to the other side, receiving terminal carries out decryption processing, recovery information after receiving information.

Claims (1)

1. but the parallel calculating method of extended precision Logistic chaos sequence is characterized in that:
Step 1, the Logistic EQUATION x n+1=μ * x n* (1-x n), x n∈ (0,1), μ ∈ (0,4), n=0,1 ..., wherein, x nBe the n time iterative value, initial parameter x 0, the μ integer quantisation is saved in one-dimension array, sets to require the precision that reaches or the number of times of specifying iteration;
Step 2, parallel algorithms first: x=x * (1-x), utilize matrix to preserve and calculate intermediate object program revises the length of dynamic array X, preserves result of calculation in dynamic array X;
Step 3, the parallel algorithms second portion: x=μ * x, utilize matrix to preserve and calculate intermediate object program, revise the length of dynamic array X, preserve result of calculation in dynamic array X;
Step 4 if precision or the iterations value of meeting the requirements of dynamically are in array X and obtain chaos sequence, is calculated and is finished; Otherwise, forward step 2 to;
In above-mentioned steps 1,
If initial parameter x 0Value be x, x is expressed as
x = Σ i = 1 n s i × D - i , D = 10 , s i ∈ { 0,1 , · · · , 9 } , s n ≠ 0 , n = 1,2 , · · ·
N is the number of significant digit after parameter value x decimal point, i.e. precision; If the value of initial parameter μ is designated as
μ = Σ i = 0 n u i × D - i , D = 10 , u i ∈ { 0,1 , · · · , 9 } , u n ≠ 0 , n = 0,1 , · · ·
The vectorial X={s of row 1, s 2..., s nBe kept in dynamic array X, dynamically first element of array X is preserved s 1, i.e. X[0]=s 1, X[1]=s 2... by that analogy; The element sum of the vectorial X of row, namely dynamically the length of array X is designated as L xThe vectorial U={u of row 1, u 2..., u nBe kept in array U;
In above-mentioned steps 2, establish y=1-xn=1-x, have
y = Σ i = 0 n y i × D - i , y i ∈ { 0,1 , · · · , 9 } , n = 1,2 , · · ·
Figure FFW00000071825700014
The vectorial Y={r of row 1, r 2..., r nBe stored in dynamic array Y; If row vector M i=r N-i+1* X, 1≤i≤n, and Mi has following form: Mi={mi 1, mi 2... mi n; Matrix M is preserved the results of intermediate calculations of x * (1-x), has following form:
M = 0 0 · · · · · · 0 0 s 1 r n s 2 r n · · · · · · s n - 1 r n s n r n 0 0 · · · · · · 0 s 1 r n - 1 s 2 r n - 1 · · · · · · s n - 1 r n - 1 s n r n - 1 0 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 0 0 s 1 r 2 s 2 r 2 · · · · · · s n - 1 r 2 s n r 2 0 · · · · · · 0 0 s 1 r 1 s 2 r 1 · · · · · · s n - 1 r 1 s n r 1 0 0 · · · · · · 0
In every row of matrix M, other elements are zero except the element of row vector M i; Note M is:
M = m 11 m 12 · · · m 1 n m 1 ( n + 1 ) · · · m 1 ( 2 n ) m 21 m 22 · · · m 2 n m 2 ( n + 1 ) · · · m 2 ( 2 n ) · · · · · · · · · · · · · · · · · · · · · m n 1 m n 2 · · · m nn m n ( n + 1 ) · · · m n ( 2 n ) n × 2 n
The initialization matrix M is null matrix: m ik=0,1≤i≤n, 1≤k≤2n; I is in matrix M for the storage line vector M, namely preserves the results of intermediate calculations of x * (1-x);
m ik＝mi j，k＝j+n-i+1，1≤i，j≤n；
Calculate this result:
α i＝(η i+d i)(mod D)，1≤i≤2n；
η i = Σ j = 1 n m ji ;
d i-1＝int[(η i+d i)/D]，1≤i≤2n；
Function int[] for getting the integer part of result of calculation, establish d 2n=0;
Row vector M i is as parallel computation in the different processors of calculating section under parallel environment independently; Supposing has N in parallel system ProsIndividual from the processor use, i the corresponding numbering of processor p i, as follows based on the parallel algorithm of Message-Passing Model:
(1) work as N Pros〉=L xThe time, master processor transmits the vectorial X of row and r iArrive from processor p iThe reception buffer memory in, vector M i is at processor p at once iMiddle calculating; After completing from processor calculating, transmission result of calculation is to master processor;
(2) work as N Pros＜L xThe time, the numbering of row vector M i and processor has following relation:
p i＝(i-1)(modN pros)+1(i＝1，2，...；p i＝1，2，...，N pros)；
I is the numbering of row vector M i; Transmit result of calculation to master processor after capable vector M i is completed in processor calculating, complete until all capable vector M i calculate;
The result of calculation of x * (1-x) is preserved in the vector T of being expert at, and form is as follows: T={ α 1, α 2... α n, α n+1... α 2n; The row vector T is kept in dynamic array Temp, and dynamically the length of array Temp is 2n; In order to carry out the calculating of next part, the length that enlarges dynamic array X is 2n, copies data element in dynamic array Temp in dynamic array X, release dynamics array Temp;
In above-mentioned steps 3, the process of parallel computation x=μ * x is except partial parameters, matrix and array length difference, and computational methods are identical with algorithm first;
In above-mentioned steps 4, setting parameter x nPrecision be L n, the precision of initial parameter μ is L u, iterations and precision have following relation:
L n+1＝2×L n+L u；L n，L u≠0；n＝0，1，...
When this iteration obtains precision that precision sets greater than the user, take the method for truncation to process, namely cast out remaining number of significant digit.
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