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LU Decomposition: Efficient Matrix Inversion and System Solving Techniques

Chapter 10 explores LU decomposition, a method that improves the efficiency of matrix inversion by separating the time-consuming elimination process from the manipulation of the right-hand side. It presents a systematic way to solve systems of equations using forward and back substitution, starting from a decomposed lower triangular matrix (L) and an upper triangular matrix (U). This method retains the same computational complexity as Gaussian elimination (in terms of floating-point operations) while significantly saving computation time. The chapter provides clear examples and explanations of this powerful technique.

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LU Decomposition: Efficient Matrix Inversion and System Solving Techniques

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  1. Chapter 10 Chapter 10

  2. LU Decomposition and Matrix InversionChapter 10 • Provides an efficient way to compute matrix inverse by separating the time consuming elimination of the Matrix [A] from manipulations of the right-hand side {B}. • Gauss elimination, in which the forward elimination comprises the bulk of the computational effort, can be implemented as an LU decomposition. Chapter 10

  3. If L- lower triangular matrix U- upper triangular matrix Then, [A]{X}={B} can be decomposed into two matrices [L] and [U] such that [L][U]=[A] [L][U]{X}={B} Similar to first phase of Gauss elimination, consider [U]{X}={D} [L]{D}={B} • [L]{D}={B} is used to generate an intermediate vector {D} by forward substitution • Then, [U]{X}={D} is used to get {X} by back substitution. Chapter 10

  4. Fig.10.1 Chapter 10

  5. LU decomposition • requires the same total FLOPS as for Gauss elimination. • Saves computing time by separating time-consuming elimination step from the manipulations of the right hand side. • Provides efficient means to compute the matrix inverse Chapter 10

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