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Thomas algorithm
1. Thomas' algorithm, also called the tridiagonal matrix algorithm (TDMA), is used to solve systems of equations with a tridiagonal matrix structure. 2. It works by applying Gaussian elimination to convert the system of equations into an upper triangular system that can then be solved using backward substitution. 3. An example problem demonstrating the algorithm is worked through, showing the conversion to upper triangular form and solution via backward substitution.
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Overview of Thomas algorithm and LU decomposition for tri-diagonal systems.
Definition of banded matrices and their characteristics including upper and lower bandwidth.
Tri-diagonal matrix specifics with non-zero elements only on the main, upper, and lower diagonals.
Explains Thomas’ algorithm used to solve tridiagonal systems using Gaussian elimination.
Mathematical formulation of tridiagonal systems and how unknowns relate to neighbouring values.
Presentation of a sample system of equations structured in matrix form for better understanding.
Describes the first stage of the algorithm focusing on modifying equations, dividing, and substituting.
Further steps in the forward elimination process addressing subsequent rows and results.
Final adjustments to the matrix after completing row operations in the forward elimination stage.
Start of backward substitution using computed values to find solutions for unknowns.
Presentation of the final values calculated for the variables x1, x2, and x3.
Reasons for the usage of Thomas algorithm including its efficiency in handling tridiagonal equations.
Discusses conditions under which the Thomas algorithm becomes unstable due to matrix singularity.
Conditions required for stability in the algorithm and techniques like pivoting to address instability.
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Thomas algorithm
- 1. Thomas Algorithm LU Decompositionfor Tri-Diagonal Systems S.K.PARIDHI
- 2. Banded matrix A bandmatrix is a sparse matrix whose non-zero entries are confined to a diagonal band, comprising the main diagonal and zero or more diagonals on either side. 1 0 0 0 2 0 0 0 3 1 4 0 6 2 5 0 7 3
- 3. The matrix canbe symmetric, having the same number of sub- and super- diagonals. If a matrix has only one sub- and one super-diagonal, we have a tridiagonal matrix etc. The number of super-diagonals is called the upper bandwidth (two in the example), and the number of sub-diagonals is the lower bandwidth (three in the example). The total number of diagonals, six in the example, is the bandwidth. Banded matrix
- 4. What is aTriDiagonal Matrix…?? A tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the first diagonal below this, and the first diagonal above the main diagonal. Considering a 4 X 4 Matrix 1 4 0 0 3 4 1 0 0 2 3 4 0 0 1 3 𝑎1 𝑏1 0 0 𝑐2 𝑎2 𝑏2 0 0 𝑐3 𝑎3 𝑏3 0 0 𝑐4 𝑎4
- 5. Tridiagonal Matrix Algorithm •Thomas’ algorithm, also called TriDiagonal Matrix Algorithm (TDMA) is essentially the result of applying Gaussian elimination to the tridiagonal system of equations. • A system of simultaneous algebraic equations with nonzero coefficients only on the main diagonal, the lower diagonal, and the upper diagonal is called a tridiagonal system of equations.
- 6. Generalizing Tridiagonal Matrix Considera tridiagonal system of N equations with N unknowns, 𝑢1, 𝑢2, 𝑢3,··· 𝑢 𝑁as given below:
- 7. Generalizing Tridiagonal Matrix TheMatrix can be written as 𝑎𝑖 𝑢𝑖−1 + 𝑏𝑖 𝑢𝑖 + 𝑐𝑖 𝑢𝑖+1 = 𝑑𝑖 Looking at the system of equations, we see that 𝑖 𝑡ℎ unknown can be expressed as a function of (𝑖 + 1) 𝑡ℎunknown. That is 𝑢𝑖 = 𝛾𝑖 𝑢𝑖+1 + 𝜌𝑖 𝑢𝑖−1 = 𝛾𝑖−1 𝑢𝑖 + 𝜌𝑖−1 𝑵𝒐𝒕𝒆: 𝑎1 = 0 & 𝑐 𝑁 = 0
- 8. Lets solve aproblem for clear understanding..!! Let us consider the system of equations 3𝑥1 − 𝑥2 + 0𝑥3 = −1 −𝑥1 + 3𝑥2 − 𝑥3 = 7 0𝑥1 − 𝑥2 + 3𝑥3 = 7 Matrix form is 3 −1 0 −1 3 −1 0 −1 3 𝑥1 𝑥2 𝑥3 = −1 7 7
- 9. STAGE I :Converting Mx = r into Ux=𝜌 • Row 1 𝟑𝒙𝒍 − 𝒙 𝟐 = −𝟏 Divide the Equation by 𝑎1, in this case 𝑎1= 3 ⇒ 𝒙 𝟏 − 𝟏 𝟑 𝒙 𝟐 = −𝟏 𝟑 Assuming the coefficient of 𝑥2 as 𝛾1 and the remaining constants as 𝜌1. Now the equations converts to, ⇒ 𝒙 𝟏 + 𝜸 𝟏 𝒙 𝟐 = 𝝆 𝟏 ⇒ 𝝆 𝟏 = − 𝟏 𝟑 , 𝜸 𝟏 = − 𝟏 𝟑
- 10. After Changing Row1 1 𝛾1 0 −1 3 −1 0 −1 3 𝑥1 𝑥2 𝑥3 = 𝜌1 7 7
- 11. Converting Mx =r into Ux=𝜌 • Row 2 Multiplying 𝒂 𝟐 (-1) in Row 1 and eliminating 𝒙 𝟏 Row 2 Row 2 −𝑥1 + 3𝑥2– 𝑥3 = 7 𝑎2 x Row 1 −𝑥1– 𝛾1 𝑥2– 0𝑥3 = −𝜌1 Subtracting 𝑥2 3 + 𝛾1 – 𝑥3 = 7 + 𝜌1 𝑥2 3 + 𝛾1 – 𝑥3 = 7 + 𝜌1 Divide by 3 + 𝛾1 , Equation becomes 𝒙 𝟐 + 𝜸 𝟐 𝒙 𝟑 = 𝝆 𝟐 𝜸2 = −1 3 + γ1 = 0.375 𝝆2 = 7 + 𝜌1 3 + γ1 = 2.5
- 12. After Changing Row2 1 𝛾1 0 0 1 𝛾2 0 −1 3 𝑥1 𝑥2 𝑥3 = 𝜌1 𝜌2 7
- 13. Converting Mx =r into Ux=𝜌 • Row 3 Multiplying 𝒂 𝟑 (-1) in Row 1 and eliminating 𝒙 𝟐 Row 3 Row 3 −𝑥2 + 3𝑥3 = 7 𝑎3 x Row 2 −𝑥2– 𝛾2 𝑥3 = −𝜌1 Subtracting 3 + 𝛾2 𝑥3 = 7 + 𝜌2 𝑥3 = 7 + 𝜌2 3 + 𝛾2 ⇒ 𝜌3 𝝆3 = 7 + 𝜌2 3 + γ2 = 3.619 ⇒ 𝑥3 = 3.619
- 14. After Changing Row3 1 𝛾1 0 0 1 𝛾2 0 0 1 𝑥1 𝑥2 𝑥3 = 𝜌1 𝜌2 𝜌3
- 15. STAGE II ->Backward Substitution 1 𝛾1 0 0 1 𝛾2 0 0 1 𝑥1 𝑥2 𝑥3 = 𝜌1 𝜌2 𝜌3 Row 2: 𝑥2 + 𝛾2 𝑥3 = 𝜌2 𝑥2 = 𝜌2 − 𝛾2 𝑥3 Substituting 𝜌2 = 2.5 𝛾2 = −0.375 𝑥2 = 3.857
- 16. STAGE II ->Backward Substitution 1 𝛾1 0 0 1 𝛾2 0 0 1 𝑥1 𝑥2 𝑥3 = 𝜌1 𝜌2 𝜌3 Row 1: 𝑥1 + 𝛾1 𝑥2 = 𝜌1 𝑥1 = 𝜌1 − 𝛾1 𝑥2 Substituting 𝜌1 = −0.333 𝛾1 = −0.333 𝑥1 = 0.952
- 17. SOLUTION 𝑥1 = 0.952 𝑥2= 3.857 𝑥3 = 3.619
- 18. Why Thomas Algorithmis used ? • This type of computation is quick. • Tridiagonal system of equations are most common. • Even most of them convert the general system of equations to tridiagonal system of equations. • It’s importance is clearly visible with a large number of unknowns.
- 19. Limitations of ThomasAlgorithm • Algorithm is unstable when 𝑏𝑖 − 𝑎𝑖 𝛾𝑖−1 = 0 • This occurs if the tridiagonal matrix is singular and in some cases non-singular also. Singular Matrix (Determinant is Zero) (*Square Matrix) (It doesn’t have a Inverse) Non-Singular Matrix (Determinant is Non-Zero) (*Square Matrix) (It has a Inverse)
- 20. Condition for algorithmto be stable 𝑏𝑖 > 𝑎𝑖 + 𝐶𝑖 for every i. What can be done when the algorithm is numerically unstable..?? Pivoting of matrix