Mathquery

              Mathematical Errors (1) Inherent Errors :             Errors which are already present in the statement of a problem before its solution , are called inherent errors . Such errors arise either due to the given data being mathematical tables , calculators or the digital computer . Inherent errors can be minimized by taking better data or by using high precision computing aids . (2) Rounding Errors :                It arises from the process of rounding off the numbers during the computation . Such errors are unavoidable in most of the calculations due to the limitations of the computing aids . Rounding errors can however be reduced by retaining at least one more significant figure at each step than that given in the data and rounding off at the last step .  (3) Truncation Errors :             ...

              Accuracy Of Numbers 1.  Approximate Numbers :            There are two types of numbers exact and approximate . Exact numbers are 2,4,9,13,7,2,6.45....etc . But there are numbers such as 4/3 =1.3333....  ,          √(2)= 1.41413 and finite number of digits . These may be approximated by numbers 1.3333, 1.4141 , 3.1416 respectively . Such numbers which represent the given numbers to certain degree of accuracy are called approximate numbers .  2. Significant Figures :                     The digits used to express a number is called significant digits (figures) . Thus each of the numbers 7845, 3.589, 0.4758 contains four significant figures while the numbers 0.00389 , 0.000587 and 0.0000296 contain only three significant figures since zeros only help to fix the position of the decimal poin...

Newton Raphson Method :        Let x₀ be an approximate root of the equation f(x) = 0 .    If x₁= x₀+h be the exact root , then f(x) = 0  ∴ Expanding f(x₀+h) by Taylor's series  we have     f(x₀+h) = f(x₀) + hf'(x₀) + h²/2 f"(x₀) +...... = 0 Since h is small , neglecting h² and higher power of h , we get          f(x₀) + hf'(x₀) = 0  ⇒    h = -f(x₀)/f'(x₀) ..........(1) Similarly starting with x₁ , a still better approximation x₂ is given by      x₂ = x₁ - f(x₁)/f'(x₁)  In general ,     xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ)       .........(2)                 where n = 0,1,2....... Which is known as                                 Newton Raphson formula or...

Backward Euler Method:        In numerical analysis and scientific computing , the backward Euler method( implicit Euler method ) is one of the most basic numerical methods for ordinary differential equations . It is similar to the standard Euler method , but differs in that it is an implicit method . 

Three_Point Gauss_Legendre Rule:        We have from  Gauss Quadrature Rule that   Rₙ₊₁(f) = w₀f(x₀) + w₁d(x₁) + ......+wₙf(xₙ) ...(1)    ⇒R₃(f) = w₀f(x₀) + w₁f(x₁) + w₂f(x₂) .....(2)                         1 Again , I(f) = ∫ f(x)dx .........(3)                        -1 Now , I(f) is given by      I(f) = R₃(f) + E₃(f) ...........(4)       where E₃(f) is its error . 

Two_Point Gauss_Legender Rule:      We know from Gauss Quadrature Rule that     Rₙ₊₁(f) = w₀f(x₀) + w₁f(x₁)+....+wₙf(xₙ) .....(1) ⇒R₂(f) = w₀f(x₀) + w₁f(x₁) ......(2)                         1 Again , I(f) = ∫ f(x) dx .........(3)                       -1 But I(f) is also given by    I(f) = R₂(f) + E₂(f) .........(4)           where E₂(f) is its error . The four unknowns w₀,w₁,x₀,x₁ are determined by using equation (2) exact for the monomials 1,x,x²,x³ i.e  E₂(xⁱ) = 0 ,  i= 0,1,2,3      ........(5) Now equation (4) can be written as   1  ∫ f(x) dx = w₀f(x₀) + w₁f(x₁) + E₂(xⁱ).....(6) -1 For f(x)= 1 :          We get from equation (6)      1     ∫ 1 dx =w...

Comparison Tests For Convergence: Comparison Test 1 (Comparison Of                                    Two Integrals) :            If f and g be two positive functions such that f(x)≤g(x) , for all x in [a,b] , then            b                                  b     (1)  ∫ f dx converges , if ∫ g dx converges           a                                   a and   b                               b    (2)  ∫ g dx diverges , if ∫ f dx diverges .          a      ...