Simultaneous Linear Equations PDF

SIMULTANEOUS LINEAR EQUATION (GAUSSIAN ELIMINATION AND JACOBI'S ITERATION METHOD) NAMES : CATHERINE CASAL KENT BRYAN PIENCENAVIES ALDREN TOLOZA XIAN CAPILITAN GAUSSIAN ELIMINATION METHOD Definitions: ❖ a popular technique for solving simultaneous algebraic equations...

SIMULTANEOUS LINEAR EQUATION (GAUSSIAN ELIMINATION AND JACOBI'S ITERATION METHOD) NAMES : CATHERINE CASAL KENT BRYAN PIENCENAVIES ALDREN TOLOZA XIAN CAPILITAN GAUSSIAN ELIMINATION METHOD Definitions: ❖ a popular technique for solving simultaneous algebraic equations Ax = b ❖ reduces the coefficient matrix into an upper triangular matrix Ux = c ❖ consists of two parts: -elimination phase -solution phase ❖ Initial Form: Ax = b ❖ Final Form: Ux = c Gauss Elimination Operations: 1. Multiplication of one equation by a non-zero constant. 2. Addition of a multiple of one equation to another equation. 3. Interchange of two equations. - Ax = b and Ux = c are equivalent if the sequence of operations produce the new system Ux = c - A is invertible if U is invertible Gauss Elimination Process: 1. Eliminate 𝑥1 from the second and third equations assuming 𝑎11 ≠ 0. 2. Eliminate 𝑥2 from the third row assuming 𝑎22 ′ ≠ 0. 3. Apply back substitution: 𝑥3 from 𝑎33 ′′ 𝑥3= 𝑏3′′ 𝑥2 from 𝑎22 ′ 𝑥2 + 𝑎23 ′ 𝑥3 = 𝑏2′ 𝑥1 from 𝑎11 𝑥1 + 𝑎12 𝑥2 + 𝑎13 𝑥3 = 𝑏1 Pivoting: Gauss elimination method fails if any one of the pivots becomes zero. ❖ What if pivot is zero? Solution: interchange the equation with its lower equations such that the pivots are not zero GAUSS-JORDAN METHOD ❖ 𝐴𝑥 = 𝑏 is reduced to a diagonal set 𝐼𝑥 = 𝑏′ where: 𝑰 = a unit matrix or identity matrix 𝑰𝒙 = 𝒃′ equivalent to 𝒙 = 𝒃′ where 𝒃′ solution vector ❖ implements the same series of row operations as implemented by Gauss except that it applies below as well as above the main diagonal - all off-diagonal elements are reduced to zero - all main diagonal elements become 1 Gauss-Jordan Process : 1. Determine if pivot is non-zero. If it is zero, swap row to succeeding rows with non-zero element. 2. Divide the pivot by itself to make the pivot equal to 1. 3. Eliminate all other elements on that column where pivot is located. 4. Go to the next row, and repeat all steps until reaching the last row. EXAMPLES : o Solve the following systems using Gaussian elimination and Gauss-Jordan process. 2𝑥 + 𝑦 − 3𝑧 = 1 4𝑥 − 2𝑦 +3 𝑧 = 8 −2𝑥 + 2𝑦 − 𝑧 = −6 Solution: GAUSSIAN ELIMINATION GAUSS-JORDAN JACOBI’S ITERATION METHOD Consider the equation: Will it converge? 3𝑥 + 1 = 0 Another iterative scheme: which can be cast into an iterative scheme as: 𝑥 = −2𝑥 − 1 𝑥+1 𝑥𝑘+1 = −2𝑥𝑘 − 1 2𝑥 = −𝑥 − 1or 𝑥 = − 2 Will this converge? which can be expressed as: 1 1 𝑥𝑘+1 = − 𝑥𝑘 − 2 2 Iterations : 𝑥𝑘 1 𝑥𝑘+1 = − − 2 2 𝑥2 1 𝑥0 1 𝑥3 = − − 𝑥1 = − − 2 2 2 2 where 𝑥0 is the initial guess 𝑥1 1 𝑥2 = − − 2 2 Jacobi’s Iteration Method ❖ aka the method of simultaneous displacements ❖ applicable to predominantly diagonal systems - Consider the system of linear equations: where 𝑎11 , 𝑎22 , and 𝑎33 are the largest coefficients - Unknowns are solved using the equations: Approximations and Iterations : ❖ Let the initial approximations be 𝑥10 , 𝑥20 , and 𝑥30 respectively, it is a general practice to assume 𝑥10 = 𝑥20 = 𝑥30 = 0 ❖ Iteration process is continued until the values of 𝑥1 , 𝑥2 and 𝑥3 are found to a pre-assigned degree of accuracy EXAMPLES : o Solve the following equations by Jacobi’s method. Solution: PRACTICE PROBLEMS : Solve the following system of equations using Gauss elimination method. 4𝑥 + 𝑦 + 2𝑧 = 11 𝑥 + 3𝑦 + 𝑧 = −4 2𝑥 + 𝑧 = 3 Solve the following system of equation using Gauss-Jordan method. 2𝑥 + 3𝑦 − 𝑧 = 10 𝑥 − 𝑦 + 𝑧 = 33 𝑦 + 2𝑧 = −11 Solve the following system below by using Jacobi’s method. 26x1 + 2x2 + 2x3 = 12.6 3x1 + 27x2 + x3 = – 14.3 2x1 + 3x2 + 17x3 = 6.0 PROBLEM 1: THANK YOU