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By Vandenbussche D., Nemhauser G. L.
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Additional resources for A branch-and-cut algorithm for nonconvex quadratic programs with box constraints
1 Least Squares Polynomial Approximations We are looking for a polynomial of degree ≤ n, p∗ (x) = p∗n xn + p∗n−1 xn−1 + · · · + p∗1 x + p∗0 that satisﬁes b a w(x) (f (x) − p∗ (x))2 dx = min b p∈Pn a w(x) (f (x) − p(x))2 dx. 1) Deﬁne f, g as b f, g = w(x)f (x)g(x)dx. a The approximation p∗ can be computed as follows: • Build a sequence (Tm ), (m ≤ n) of polynomials such that (Tm ) is of degree m, and such that Ti , Tj = 0 for i = j. 2) n ai Ti . i=0 The proof is rather obvious and can be found in most textbooks on numerical analysis .
The conditions “2a ≥ r + 1” and “a ≤ r − 1” cannot be simultaneously satisﬁed in radix 2. Nevertheless, it is possible to perform parallel, carry-free additions in radix 2 with digits equal to −1, 0, or 1, by using another algorithm, also due to Avizienis (or by using the borrow-save adder presented in the following). 3 presents an example of the execution of Avizienis’ algorithm in the case r = 10, a = 6. Redundant number systems are used in many instances: recoding of multipliers, quotients in division and division-like operations, on-line arithmetic , etc.
The exponent used for representing zero). 2). 94065645841246544 × 10−324 . In a ﬂoating-point system with correct rounding and subnormal numbers, the following theorem holds. Theorem 2 (Sterbenz Lemma) In a ﬂoating-point system with correct rounding and subnormal numbers, if x and y are ﬂoating-point numbers such that x/2 ≤ y ≤ 2x, then x − y will be computed exactly. That result is useful when computing accurate error bounds for some elementary function algorithms. The IEEE-754 standard also deﬁnes special representations for exceptions: • NaN √ (Not a Number) is the result of an invalid arithmetic operation such as −5, ∞/∞, +∞ + (−∞), .
A branch-and-cut algorithm for nonconvex quadratic programs with box constraints by Vandenbussche D., Nemhauser G. L.