D. Knuth. Seminumerical Algorithms. Addison-Wesley, Reading, MA, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The grammar based compression is an optimization problem, given an input string, to nd a small context free grammar which generates the single string. This problem is known to be NP hard and not approximable within a constant factor [9] and due to a relation with an algebraic problem [6], it is unlikely to found an algorithm approximating this problem within O(log n= log log n) The framework of the grammar based compression can uniformly describe the dictionarybased coding schemes which are widely presented for real world text compression. For example, LZ78 [16] including LZW ....

D. Knuth. Seminumerical Algorithms. Addison-Wesley, 441-462, 1981.

....we can represent the CA rule as a polynomial [8] P (x) f 1 f 2 x f 3 x Delta Delta Delta f 2r 1 x 2r 1 after separating out the constant f 0 . We can think of P (x) as the tth row of a Green s function for the CA, and by using fast algorithms to raise P (x) to the tth power [9], we can predict the CA in O(t log t) in serial (on a random access machine) or O(log t) in parallel [1] We now show that a non autonomous version of (1) where the f i vary in space and time, is still predictable in O(log t) parallel time. Definition. NC k is the class of problems that ....

D.E. Knuth, Seminumerical Algorithms. Addison-Wesley, 1981.

....size 2 so that each byte in the data item corresponds to a single y coordinate. This decision limits us to 256 unique x coordinates (i.e. N 256) Since the polynomial is interpolated at the x coordinate points many times, interpolation coefficients are precalculated using Newton s Formula [27]. Given these m (N m) constants, each of the N m code fragment interpolations require m multiplications and 2m additions. Multiplication in the Galois Field is implemented as a lookup table (i.e. a 2 2 B= 64 KB lookup table is used) Addition in the Galois Field is just bitwise XOR. ....

Donald Ervin Knuth. Seminumerical algorithms, volume 2. Addison-Wesley, 1981.

....arithmetic see [2] 8.9) For a constant r3, a division with remainder can be computed with r3M(n) operations in Fq. Finally, we need the computation of h q rood f for polynomials h and f of degree at most n. This exponentiation can be done by means of the classical repeated squaring method (see [17], p. 461 462) In this case, the number of products needed is Cq = 1og 2 q v(q) 1, with v(q) the number of ones in the binary representation of q. Therefore, the cost of computing h q rood f by this method is rCqM(n) operations in Fq for both arithmetics. For an excellent reference book in the ....

D.E. KNUTH. The art of computer programming, vol. 2: seminumerical algo- rithms. Addison-Wesley, Reading MA, 3 edition, 1997.

....capable of inverting some imperative algorithms, but are inadequate for most general cases. In fact, the abductive modeling of while loops is only an aid in inverting loops and is not strictly required. Other imperative algorithms, including Fibonacci numbers, bubble sort, and Knuth s Algorithm P [Knu81], have been successfully inverted with the use of more advanced logic programming control strategies. Abstract interpretation, dynamic control mechanisms such as coroutining and intelligent backtracking, and search heuristics are very useful in this regard. One promising avenue being investigated ....

Donald E. Knuth. Seminumerical Algorithms. Addison-Wesley, 2nd edition, 1981.

....bit6 bit 8 . bit30 bit31 bit1 bit2 Direction of shift bit7 Figure 1: Logical Feedback Shift Register Random Number Generator The random number is read from the highest bits as required. The obvious weakness of this type of RNG is that sequential values fail the serial test described by Knuth [6]. At any time step t there is a 50 probability that the value at time t 1 can be predicted. If for an LFSR of length n at time t the value is v, then at time t 1 the value will be v 2 or v . This is shown in Figure 2 where pairs of values v t and v t 1 are plotted. It can ....

E. Knuth, Donald. Semi numerical algorithms, volume 2. Addison-Wesley Publishing Company, 1969.

....from an external source, such as a time of day clock, or other source of noise. It also allows the RNG to be preset to a known seed for producing repeatable results. There is a suspicion that this RNG is not ideal because LFSRs are known to perform poorly in the serial test described by Knuth [10] and this is an area for further investigation. 5. Breeding Policy and Operators To conserve memory, a steady state breeding policy was used. Tournament selection is used with a tournament size of two. Larger tournament selection makes little sense with very small populations. The operators ....

E. Knuth, Donald. Semi Numerical Algorithms, volume 2. Addison-Wesley Publishing Company, 1969.

....is discarded without having been used. This should also be done for the Oswald Aigner algorithm. In addition, an extra double needs to be performed to convert each DAAD into DADAD, and the double s output is likewise ignored. Alternatives randomized algorithms exist. Standard m ary exponentiation [5] is not subject to this type of attack, but re use of operands might be detected [16] making that method unsuitable even with key blinding in place. The MIST algorithm [17 19] and an overlapping windows method [3] currently seem to be the most robust choices under these types of attack. 5 ....

D. E. Knuth, The Art of Computer Programming, vol. 2, "Seminumerical Algorithms", 2nd Edition, Addison-Wesley, 1981, 441-466.

....a constant factor, approximating the smallest grammar for a string is at least as hard as approximating the shortest addition chain containing a speci ed set of numbers. 33 3.3 Background on Addition Chains Addition chains have been studied extensively for decades. There is a survey in Knuth [19] and a less comprehensive, but more recent survey due to Thurber [36] The classical addition chain problem is to nd the shortest addition chain containing a single, speci ed integer n. This is closely allied with the problem of nding the smallest grammar for the unary string x . All major ....

Donald E. Knuth. Seminumerical Algorithms. Addison-Wesley, 1981.

....fields (with q = 2 1 as a Mersenne number) and fields of characteristic 2, i.e. q = 2 . Large prime fields: There is a good reason to choose q as a Mersenne number. No integer division is required for modular reduction in modular multiplication modulo a Mersenne number q = 2 1, see [23] [9]. Suppose a, b, t, u F q , and c = ab = 2 t u, we have c = t u) mod q. There is no Mersenne number between 2 and 2 . Therefore, ECC cannot take advantage of the shortcut for modular multiplication modulo a Mersenne number, when 2 . However, things are di#erent for HCC ....

D.E. Knuth, and E. Donald E., Seminumerical Algorithms, Addison-Wesley, 1981.

....that i ,i ,k = l A l A s i ,i , j mod p where the modulus in the numerator is taken balanced. The only problem could be that p were too small to capture l A the numerator of s i ,i ,k . But the integral coefficients of factors of det(A ) are absolutely bounded by 2 B (see [14], Section 4.6.2, Exercise 20) Now clearly 2 l A 2 B p and we have the following theorem. Theorem 4.1: SMITH FORM over Q[x ] is in P. 5. Rational Canonical form and Similarity If A is a matrix over a field F , then the diagonal entries of the Smith normal form of xI A (over F [x ] are ....

Knuth, D.E., The Art of Programming, Vol. 2, Seminumerical Algorithms, 2nd edition, Addison Wesley, 1981.

....the case of data transferred, continuous in the non negative integers. As such the values of the variables do not naturally fall into a finite number of categories, which makes using the well known chi squared test less than ideal because it requires somewhat arbitrary choices regarding binning [Knuth81, DS86] The goodness of fit test commonly used with continuous data is the Kolmogorov Smirnov test. The authors of [DS86] however, recommend the Anderson Darling ( test [AD54] instead. They state that is often much more powerful than either Kolmogorov Smirnov or chi squared, and that ....

D. Knuth, "Seminumerical Algorithms", Second Edition, Addison-Wesley, 1981.

....of the shortest addition chain for r An addition chain for r is a list of positive integers r# such that, for each i 1, there is some j and k with 1 k iand a i a j a k . A short addition chain for r gives a fast algorithm for computing g compute g a l 1 . See Knuth [13] for an excellent introduction to addition chains. Let lr be the length of the shortest addition chain for r. The exact value of lr is known only for relatively small values of r. It is known that, for r large, lr=log r 1 o1 # (1) The lower bound was shown by Erd os [11] using a counting ....

....the sequence that maps to (3) is 1# 2# 3# 4# 7# 8# 15# 23# Downey et al. 10] showed that the problem of finding the shortest addition sequence is NP complete. 2. BASIC METHODS 2.1. Binary Method This method is also known as the square and multiply method. It is over 2000 years old; Knuth [13] discusses its history and gives references. The basic idea is to compute g using the binary expansion of r.Let r= Then the following algorithm will compute g if c d 1 then a return a. At each step of the for loop a is equal to g s , where the binary representation of s is a ....

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D. E. Knuth, "Seminumerical Algorithms," 2nd ed., "The Art of Computer Programming," Vol. 2, Addison-Wesley, Reading, MA, 1981.

....speed up the decryption and signature operation by application of the Chinese Remainder Theorem. In this case the decryption of the ciphertext C works like this: M q = C dq mod q M p = C dp mod p M p = M p ,M q mod p M = #M p # invq mod p# q M q We u se an m bit window method (see e.g. [16]) for exponentiation, where l is the bitlength of the exponent. On average we will need about 2 m,1 2 m , 1 2 m dl=me l (1) modular multiplications. Thus we may optimally choose m =5for bitlength up to 512 and m =6for bitlenght 768 and 1024. Therefore we need about 2 # 628 2 = 1258 ....

....on the harddrive. 4. Secure Single Login DL based (without Precomputation) This variant also uses the SSLogin mechanism discussed in section 5, where the temporary key pairs are ElGamal for encryption and DSA for signatures. All exponentiations are performed using the (ordinary) window method [16] with a suitable window size depending on the bitlength of the exponent. 4.1 Security Since the (long term) secret keys are only exposed until the temporary certificates are signed, the security is high. 4.2 Usability Since the user only has to enter the PIN once, the usability may be ....

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D.E. Knuth: The Art of Computer Programming Vol. 2: Seminumerical algorithms, Addison-Wesley, Reading MA, 1981

....integers. How many multiplications are required to compute x k1 ; x k2 ; x kp (Algorithms that use other operations are ruled out. Thus, more precisely, what is the shortest addition chain containing all of k 1 ; k 2 ; k p The theory of addition chains is extensive [8]. This problem is known to be NP hard if the integers k i are given in binary [5] However, even if they are written in unary, apparently no polynomial time algorithm with approximation ratio o(log n= log log n) is known, where n = P k i . The following theorem states that improving the ....

D. Knuth. Seminumerical Algorithms. Addison-Wesley, 1981, pp. 441-462.

....factoring 384 80 512 1656 1024 513 752 GPS scheme [PS98] n = 1024 Discrete log. modulo n 384 80 1024 1796 3072 1024 1264 Table 1. Performance of signature schemes Table 1 gives the performance of various signature schemes including ours. Here, a primitive arithmetic of binary methods [Knu81] is used. For all schemes in the table, we set up the parameter under the line of the one key attack scenario. Hence the size of secret key in GPS scheme is 1024 bits. For more discussion on it, we refer to [Po00] UMP means the underlying mathematical problem that the signature scheme relies on ....

D. E. Knuth: "Seminumerical Algorithms", The art of computer programming, vol.2, Second edition, Addison-Wesley, 1981.

.... of random numbers as can be seen in Figure 1(a) Also there exists a value (more precisely 0) into which the generator is driven and where it gets stuck [1] A little bit better example is the generator x n 1 = 8121x n 28411) mod 134456) 39) which has been shown to be the optimal choice [7] for a 32 bit integer case with the first bit reserved as the sign bit. There are only 134456 possible values and there is some correlation between the the consequent values so the generator is not an ideal one but, as can be seen on the Figure 1(d) the points are much more evenly distributed ....

D.E. Knuth, Semi-Numerical Algorithms, 2 nd Edition, The Art of Computer Programming, Vol. 2, Addison-Wesley (1981).

....a flat probability distribution in the allowed range. Another function returning a random floating point number with a gaussian probability distribution (zero mean and unit variance, or user supplied variance) Random sequences are generated using the subtractive algorithm, originally due to Knuth [1981]; our implementation, however, is based on the description given by Press et al. 1992] This algorithm was chosen because it has a very long period (necessary for simulating low bit error rates) and also because it does not suffer from low order correlations, simplifying the generation of random ....

KNUTH, DONALD E. 1981. Seminumerical Algorithms. Second edn. The Art of Computer Programming, vol. 2. Addison Wesley.

....# ) 1 # # # ### I[ # # # # # # ]# (2. 1) The Gaussian errors have been generated using the Box Muller method, which uses uniform deviates randomly drawn from two di erent random number generators, the congruential random number generator with long period proposed by L Ecuyer (1988) and Knuth s (1998) subtractive method, initialized with two di erent seeds. An important pointisthechoice of the number of draws # from the bootstrap DGP, as it involves a trade o between power loss and computing time. Given that the computation of the 4 CVM and NCVM bootstrap tests involve numerical ....

Knuth, D. E. (1998). ### ### ## ######## ###########,volume 2, Seminumerical Algorithms. Addison Wesley.

....separate Poly k from PiecePoly k ; since it has an upper bound of O(nd log jk) where j is the number of components of each map [13] branching could conceivably be simulated by polynomials of degree jk. Finally, we note that combining the above with results of Cucker and Grigoriev [9] and Koiran [19, 20] give us bounds on Poly and NPoly tighter than theorem 10, as well as bounds on PieceLin(R) and NPieceLin(R) Recall [30] that a machine has polynomial advice if it has access to an oracle whose advice depends only on the length of the input, and is polynomially long. Classes with polynomial ....

D.E. Knuth, Seminumerical Algorithms. Addison-Wesley, 1981. 36

....from LFSRs are fairly random. This is not surprising because single bit shift register sequences demonstrate randomness properties [12] Counting Two State Transition Tour Sequences 12 This is generally is not true for other pseudorandom generators like linear congruential (LC) generators [13]. For example, if the bit sequence is derived from the least significant bit of the word generated by the LC generator then the sequence will be alternating 0 s and 1 s. This is hardly random. Figure 7 clearly shows the departure of LFSR and LC generators from pure random pattern generators. The ....

Knuth, D.E., Seminumerical Algorithms, Addison-Wesley Publishing Company, Inc., Reading, Mass., 1969.

....three kinds of these: 1. One can choose the curve, and the base field over which it is defined, so as to optimize the efficiency of elliptic scalar multiplication. Thus, for example, one might choose the field of integers modulo a Mersenne prime, since modular reduction is particularly efficient [Knu81] in that case. This option is not available for, say, RSA systems, since the secret primes are chosen randomly in order to maintain the security of the system. 2. One can use the fact that subtraction of points on an elliptic curve is just as efficient as addition. The analogous statement for ....

D. E. Knuth, Seminumerical Algorithms, Addison-Wesley, 1981, p. 272.

....from a uniform distribution over the entire range of the attribute. It is to be noted that long sequences of uniform random numbers generated by typical Unix random number generators (Linear Congruential Generators) have been shown to exhibit regular behavior by falling into specific planes [Knu80] To avoid this, we use a random number generator called the Inversive Congruential Generator which has better statistical properties [EHG92] After populating the user defined cluster we add an additional 10 noise records to the data set. Values for all the attributes in these noise records are ....

Donald E Knuth. The Art of Computing, volume 2, Seminumerical Algorithms. Addison Wesley, 3 rd edition, 1980.

....= U (ff)A(ff) A(ff) U (ff) when a 6= 0 21 in which U and V in the last rule follow from Bezout s equality: since a 6= 0, A and OE are coprime, as proved just below ( Then after Bezout s theorem, there exist U and V such that U (x)A(x) V (x)OE(x) 1. The subresultant PRS algorithm [Knu81] is a good way to compute U and V . The last thing to prove K(ff) is computable is the computability of the equality test between two numbers b = B(ff) and c = C(ff) which boils down to the nullity test of their difference a = b Gamma c = A(ff) If OE was irreducible, A(ff) 0 would be ....

D.E. Knuth. Seminumerical Algorithms, volume 2. Addison-Wesley, Reading, Mass., 1981.

.... 2,2 s 4,6 s 28 s 297 s (Karat) 5,1 s 33.6 s 1,8 ms 138 ms 3,9 s 32 s 1,8 ms 177 ms = 8,6 s 68,1 s 3,8 ms 380 ms exp 534 s 5,7 ms 690 ms 325 s sin 194 s 2,2 ms 240 ms 115 s The algorithms for exp and sin are the (almost) immediate ones using Taylor series, division is implemented as in [Kn81]. The next table compares timings for a 50 bit division in the multiple precision package and the corresponding floating point arithmetic on certain computers (in a context of a loop taking arguments from an array and storing them again in the array) The internal word size is the number of ....

Knuth, D. E., The art of computer programming, Vol. 2 (Semi numerical algorithms), (Addison-Wesley Publishing Company 1981)

....of size N . Since the graph in question consists of the n sites of the cluster itself, our algorithm needs O(n 3 ) processors to carry out pebble or hole sweeps in polylogarithmic time. In fact, we can reduce this somewhat by using more sophisticated methods for parallel circuit multiplication [13]. 4.2 Simulations of the relaxation algorithm Our algorithm consists of alternating pebble sweeps and hole sweeps until the energy is zero, at which point the cluster is in the correct con guration. How many steps are required to do this To explore this question we simulated the relaxation ....

D.E. Knuth, Seminumerical Algorithms. Addison-Wesley, 1981.

....random from a uniform distribution over the entire range of the attribute. It is to be noted that long sequences of uniform random numbers generated by typical Unix random number generators (Linear Congruential Generators) have been shown to exhibit regular behavior by falling into specific planes [19]. To avoid this, we use a random number generator called the Inversive Congruential Generator which has better statistical properties [20] After populating the user defined cluster we add an additional 10 noise records to the data set. Values for all the attributes in these noise records are ....

Donald E Knuth. The Art of Computing, volume 2, Seminumerical Algorithms. Addison Wesley, 3 rd edition, 1980.

....3kva; floor area: 8 square meters. Input: photoelectric paper tape reader; output: teletype. The PC 1 seemed to be the first computer that installed the interruption mechanism, which enabled us to experiment with multiple programming in 1959. The research on modular computations referred to in [2] was conducted on this machine. The present paper reviews the parametron circuits and memory structure in sections 1 and 2, then the structure of memory and registers in section 3. In section 4, the teletype code used by the PC 1 is given so that the readers can understand the details of the ....

D. Knuth: The Art of Computer Programming, vol 2, Seminumerical Algorithms, 3rd ed. p. 291.

....the decryption and signature operation by application of the Chinese Remainder Theorem. In this case the decryption of the ciphertext C works like this: M q = C dq mod q M p = C dp mod p M p = M p Gamma M q mod p M = M p Delta invq mod p) q M q We use an m bit window method (see e.g. [16]) for exponentiation, where l is the bitlength of the exponent. On average we will need about 2 m Gamma1 2 m Gamma 1 2 m dl=me l (1) modular multiplications. Thus we may optimally choose m = 5 for bitlength up to 512 and m = 6 for bitlenght 768 and 1024. Therefore we need about 2 ....

....on the harddrive. 4. Secure Single Login DL based (without Precomputation) This variant also uses the SSLogin mechanism discussed in section 5, where the temporary key pairs are ElGamal for encryption and DSA for signatures. All exponentiations are performed using the (ordinary) window method [16] with a suitable window size depending on the bitlength of the exponent. 4.1 Security Since the (long term) secret keys are only exposed until the temporary certificates are signed, the security is high. 4.2 Usability Since the user only has to enter the PIN once, the usability may be ....

[Article contains additional citation context not shown here]

D.E. Knuth: The Art of Computer Programming Vol. 2: Seminumerical algorithms, Addison-Wesley, Reading MA, 1981

....table of such numbers generated in advance. Without such aids, which are either impractical or not generally available, the alternative is to use numerical algorithms. No deterministic algorithm can produce a sequence of numbers that would have all of the properties of a truly random sequence [3]. However, for all practical purposes it is only necessary that the numbers produced appear random, i.e. pass certain statistical tests for randomness. Although these generators produce pseudo random numbers, we continue to call the random number generators. The starting point for generating ....

...., where the period is 2 24, and the initial seed must be odd. The Uniform( method uses the linear congruential generator (seed is LCGSeed) based on the algorithm in [4] and the results of this are shuffled with the multiplicative generator (see is MGSeed) as suggested by Maclaren and Marsaglia [3], to obtain a sufficiently uniform random distribution, which is then returned. 12 Thanks to Professor I. Mitrani for his help in developing this. The C SIM User s Manual D str but on Funct ons 23 The Error( method returns a chi square error measure on the uniform distribution function. By ....

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Knuth Vol2, "Seminumerical Algorithms", Addison-Wesley, 1969, p. 117.

....of primes, and discuss parameter choices which are particularly well suited for machine implementation. Keywords: modular arithmetic, elliptic curves. Introduction It has long been known that certain integers are particularly well suited for modular reduction. The best known examples (e.g. [1]) are the Mersenne numbers m = 2 k Gamma 1. In this case, the integers (mod m) are represented as k bit integers. When performing modular multiplication, one carries out an integer multiplication followed by a modular reduction. One thus has the problem of reducing modulo m a 2k bit number. ....

Knuth, Donald E., Seminumerical Algorithms, Addison-Wesley, 1981.

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D. Knuth. Seminumerical Algorithms. Addison-Wesley, Reading, MA, 1997.

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D.E. Knuth, Seminumerical algorithms, Addison-Wesley, 1981.

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D.E. Knuth: The Art of Computer Programming Vol. 2: Seminumerical algorithms, Addison-Wesley, Reading MA, 1981

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D. Knuth. The Art of Computer Programming, volume 2, "Seminumerical Algorithms". Addison Wesley, Reading, MA, third edition edition, 1998.

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Knuth, D., Semi-Numerical Algorithm, The Art of Computer Programming, AddisonWesley, Second Edition, Vol. 2, 1981.

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Donald E. Knuth. The Art of Computer Programming, volume 2 (Semi-Numerical Algorithms). Addison-Wesley, Reading, Massachusets, 1981.

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D. E. Knuth. Seminumerical Algorithms. AddisonWesley, 1969.

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D.E. Knuth (1981), Seminumerical Algorithms (2nd ed.), Addison Wesley, Reading, MA.

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Knuth, D.: Seminumerical algorithms. 2nd ed. Reading, MA: Addison-Wesley 1981. Gelfand, A.E.: On the cyclic behavior of random transformations on a finite set. Tech. rept. 305, Dept. of Statistics, Stanford Univ. (August 1981).

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Knuth DE: The Art of Computer Programming, Volume 2, Seminumerical Algorithms. Addison--Weseley, Reading, MA, Third Edition, 1997.

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D. Knuth, "Seminumerical algorithm," The Art of Computer Programming vol. 2, 1997.

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D. Knuth, Seminumerical Algorithms, The Art of

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D.E. Knuth, Seminumerical Algorithms. Addison-Wesley, 1981.

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E. Knuth, Donald. Semi numerical algorithms, volume 2. Addison-Wesley Publishing Company, 1969.

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Knuth, D. E.: The art of computer programming, seminu- merical algorithms. Vol. 2. 2rid ed. Massachusetts: Reading, 1981.

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D.E. Knuth, "Seminumerical algorithms," in The Art of Computer Programming. Reading, MA: Addison-Wesley, 1969, vol. 2.

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D.E. Knuth, "Seminumerical algorithms", 2nd ed., vol. 2 of The art of computer programming (Reading Mass.: Addison Wesley), 1981.

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D. E. Knuth, "Seminumerical Algorithms", Addison-Wesley (1981).

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