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The chakravala method (Sanskrit: चक्रवाल विधि) is a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation.It is commonly attributed to Bhāskara II, (c. 1114 – 1185 CE) [1] [2] although some attribute it to Jayadeva (c. 950 ~ 1000 CE). [3]
Two other solutions are x = 3, y = 6, z = 1, and x = 8, y = 9, z = 2. There is a unique plane in three-dimensional space which passes through the three points with these coordinates, and this plane is the set of all points whose coordinates are solutions of the equation.
Pólya mentions that there are many reasonable ways to solve problems. [3] The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included: Guess and check [9] Make an orderly list [10] Eliminate possibilities [11] Use symmetry [12]
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm that, for any given Diophantine equation (a polynomial equation with integer coefficients and a finite number of unknowns), can decide whether the equation has a solution with all unknowns taking integer values.
The following scores (in addition to 1, 2, and 4) cannot be made from multiples of 5 and 7 and so are almost never seen in sevens: 3, 6, 8, 9, 11, 13, 16, 18 and 23. By way of example, none of these scores was recorded in any game in the 2014-15 Sevens World Series .
For example, taking the statement x + 1 = 0, if x is substituted with 1, this implies 1 + 1 = 2 = 0, which is false, which implies that if x + 1 = 0 then x cannot be 1. If x and y are integers, rationals, or real numbers, then xy = 0 implies x = 0 or y = 0. Consider abc = 0. Then, substituting a for x and bc for y, we learn a = 0 or bc = 0.
For solving the cubic equation x 3 + m 2 x = n where n > 0, Omar Khayyám constructed the parabola y = x 2 /m, the circle that has as a diameter the line segment [0, n/m 2] on the positive x-axis, and a vertical line through the point where the circle and the parabola intersect above the x-axis.
The standard convergence condition (for any iterative method) is when the spectral radius of the iteration matrix is less than 1: ρ ( D − 1 ( L + U ) ) < 1. {\displaystyle \rho (D^{-1}(L+U))<1.} A sufficient (but not necessary) condition for the method to converge is that the matrix A is strictly or irreducibly diagonally dominant .