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An Interval Branch-and-Bound Algorithm based on the General Quadratic Form 173⎧⎪⎨with⎪⎩A = 0 ,b =(0, · · · , 0, xL − x U2c = xL + x U,2e + = 0,e − = 0,e = 0.), 0, · · · , 0 ,The rank i where b i = xL − x Uis generally determined by the variable x. When a vector2is considered x ∈ IR n , then the rank i corresponds to x i .General Quadratic Form −→ Interval:̂x = ɛ T Aɛ + b T ɛ + c + e + ɛ n+1 + e − ɛ n+2 + eɛ n+3=n∑n∑a ij ɛ i ɛ j + b i ɛ i + c + e + ɛ n+1 + e − ɛ n+2 + eɛ n+3i,j=1i=1−→ X =n∑[−|a ij |, |a ij |] +i,j=1i≠jn∑[−|b i |, |b i |] +i=1[0, e + ] + [−e − , 0] + [−e, e].n∑a ii [0, 1] + [|c|, |c|] +i=1Other conversions, such as for example between general quadratic forms and affine forms,are possible, refer to [10].The operations between general quadratic forms are performed as follows:let ̂x and ŷ be two general quadratic forms and a a real number, witĥx = ɛ T A x ɛ + b T x ɛ + c x + e + x ɛ n+1 + e − x ɛ n+2 + e x ɛ n+3 ,ŷ = ɛ T A y ɛ + b T y ɛ + c y + e + y ɛ n+1 + e − y ɛ n+2 + e y ɛ n+3 .One obtains:−bx = −ɛ T A xɛ − b T x ɛ − c x + e − x ɛ n+1 + e + x ɛ n+2 + e xɛ n+3,bx + by = ɛ T (A x + A y)ɛ + (b x + b y) T ɛ + (c x + c y) + (e + x + e + y )ɛ n+1 + (e − x + e − y )ɛ n+2 + (e x + e y)ɛ n+3,bx − by = ɛ T (A x − A y)ɛ + (b x − b y) T ɛ + (c x − c y) + (e + x + e − y )ɛ n+1 + (e − x + e + y )ɛ n+2 − (e x + e y)ɛ n+3,bx ± a = ɛ T A xɛ + b T x ɛ + (c x ± a) + e + x ɛ n+1 + e − x ɛ n+2 + e xɛ n+3,( ɛ T (a × A x)ɛ + (a × b x) T ɛ + a × c x + a × e + x ɛ n+1 + a × e − x ɛ n+2 + a × e xɛ n+3, if a > 0,a × bx =ɛ T (a × A x)ɛ + (a × b x) T ɛ + a × c x + |a| × e − x ɛ n+1 + |a| × e + x ɛ n+2 + |a| × e xɛ n+3, if a < 0.For the multiplication (a non-affine operation), the following computations are performed:̂x × ŷ = ɛ T A × ɛ + b T ×ɛ + c × + e + × ɛ n+1 + e − × ɛ n+2 + e × ɛ n+3 , (3)⎧⎨ A × = c y A x + c x A y + b x b T y ,with b⎩ × = c y b x + c x b y ,c × = c x c y .The computations of the errors e + , e − , e are not so obvious, but are quiet difficult and attentionmust be paid on it, see [10].Therefore, this technique allows to construct a new inclusion function based on this generalquadratic form; i.e. one converts all interval vector with dimension n into n general quadratic