Numerical Solutions of Transcendental Equations and System of Linear Equations and Expansion of Function.
- What is a transcendental equation and how do I locate an interval containing its root? In one line A transcendental equation contains non-polynomial functions such as e^x , log x , sin x (e.g. x=cos x ); we locate a root by fin... 1
- How does the Newton–Raphson method find a root of f(x) = 0 and why does it work? In one line Newton–Raphson improves a guess x_n by sliding down the tangent line to the x -axis: x_(n+1)=x_n-f(x_n)/f'(x_n) . Explained simp... 1
- How does the Regula Falsi (false position) method find a root, and how is it different from Newton–Raphson? In one line Regula Falsi joins the points (a,f(a)) and (b,f(b)) on opposite sides of the axis by a straight chord and takes where the chord... 1
- How do I derive Newton–Raphson iterative formulas for square roots, cube roots and reciprocals? In one line Write the required number as the root of a simple equation, such as x²-N=0 for sqrt N , and apply x_(n+1)=x_n-f(x_n)/f'(x_n) to... 1
- How does the Gauss–Jacobi iteration method solve a system of linear equations? In one line Gauss–Jacobi rewrites each equation to make one unknown the subject, then repeatedly substitutes the previous round's values int... 1
- How is the Gauss–Seidel method different from Gauss–Jacobi, and why does it converge faster? In one line Gauss–Seidel is Jacobi with one change: each newly computed value is used immediately in the following equations of the same ite... 1
- What is diagonal dominance and why must equations be rearranged before Jacobi or Seidel iteration? In one line A system is (strictly) diagonally dominant when, in every row, the size of the diagonal coefficient is greater than the sum of t... 1
- What are indeterminate forms 0/0 and ∞/∞, and how does L'Hospital's rule evaluate them? In one line When a limit takes the form 0/0 or frac∞∞ , L'Hospital's rule says limf(x)/g(x)=limf'(x)/g'(x) , provided the new limit exists.... 1
- How do I evaluate other indeterminate forms like 0·∞, ∞−∞, 1^∞, 0⁰ and ∞⁰? In one line Convert them to 0/0 or frac∞∞ : rewrite products and differences as a single fraction, and for powers take logarithms first; the... 1
- How does Gauss elimination solve a system of linear equations step by step? In one line Gauss elimination uses row operations to turn the augmented matrix into upper-triangular form, then finds the unknowns by back-s... 1
- How does the Gauss–Jordan method solve linear equations and how is it different from Gauss elimination? In one line Gauss–Jordan continues the elimination above the pivots as well, reducing the coefficient matrix to the identity so that the sol... 1
- How do I expand a function as a Taylor or Maclaurin series in powers of x or (x − a)? In one line Taylor's series writes f(x) as f(a)+(x-a)f'(a)+(x-a)²/2!f''(a)+dots ; Maclaurin's series is the special case a=0 . Explained sim... 1
- Apply Gauss-Seidel iteration method to solve the following linear equations up to the three iterations. \(20x+y-2z=17,\ 3x+20y-z=-18,\ 2x-3y+20z=25\). The system is diagonally dominant. x=frac120(17-y+2z), y=frac120(-18-3x+z), z=frac120(25-2x+3y) . Start y=z=0 , using latest values. Iterati... 1
- Using Newton-Raphson Method, find the real root of \(x^3-2x-5=0\) correct to three decimal places. f(x)=x³-2x-5,\ f'(x)=3x²-2 . f(2)=-1<0,\ f(3)=16>0 : root lies in (2, 3), nearer 2. Take x₀=2 . x_(n+1)=x_n-x_n³-2x_n-5/3x_n²-2=2x_n³+5/3x_n... 1
- Solve \(10x+y+z=12,\ x+10y+z=12,\ x+y+10z=12\) using the Gauss-Jacobi's method by taking two iterations. Diagonally dominant. x=12-y-z/10,\ y=12-x-z/10,\ z=12-x-y/10 . Start x₀=y₀=z₀=0 . Iteration 1: x₁=y₁=z₁=12/10=1.2 Iteration 2: x₂=12-1.2-1.2... 1
- Using the Newton-Raphson method find the root of the equation \(2x^3-3x+4=0\) lying between -2 and -1, correct to four places of decimals. f(x)=2x³-3x+4,\ f'(x)=6x²-3 . f(-2)=-6,\ f(-1)=5 → root in (−2, −1); f(-1.5)=1.75 , so take x₀=-1.5 . x_(n+1)=x_n-2x_n³-3x_n+4/6x_n²-3=4x_n³... 1
- Solve the following equations by Gauss-Seidel method by taking two iterations. \(28x+4y-z=32,\ 2x+17y+4z=35,\ x+3y+10z=24\). Diagonally dominant. x=frac128(32-4y+z),\ y=frac117(35-2x-4z),\ z=frac110(24-x-3y) . Start y=z=0 , using latest values. Iteration 1: x₁=32/2... 1
- Using Newton Raphson's Method, find an iterative formula for \(\sqrt[3]{150}\). Let x=sqrt[3]150⇒ f(x)=x³-150=0,\ f'(x)=3x² . Newton–Raphson: x_(n+1)=x_n-f(x_n)/f'(x_n)=x_n-x_n³-150/3x_n² Iterative formula: x_(n+1)=2x_n³... 1
- Apply the Gauss Seidel method, to solve the following system of linear equations up to the two iterations. \(15x+3y-2z=85,\ x-2y+8z=5,\ 2x+10y+z=51\). Rearrange for diagonal dominance: 15x+3y-2z=85 ; 2x+10y+z=51 ; x-2y+8z=5 . x=1/15(85-3y+2z), y=1/10(51-2x-z), z=1/8(5-x+2y) Start with y=z=0... 1
- Find the three sets of iterative solutions of the following equations by Gauss Jacobi method: \(5x-y+z=10;\ 2x+4y=12;\ x+y+5z=-1\). The system is diagonally dominant ( 5>1+1,\ 4>2,\ 5>1+1 ). x=1/5(10+y-z), y=1/4(12-2x), z=1/5(-1-x-y) Start with x₀=y₀=z₀=0 ; in Jacobi's me... 1