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Matrix Inversion Using Cholesky Decomposition
Aravindh Krishnamoorthy, Deepak Menon
TL;DR
Matrix inversion methods based on Cholesky decomposition use equation solving or triangular operations, motivating a reduction in computational work. The paper proposes avoiding an intermediate solution and reports 16-17% fewer operations, with fixed-point simulations indicating good numerical accuracy and the lowest average norm error among the compared methods.
Problem
Existing Cholesky-based inversion algorithms use equation solving or triangular matrix operations, leaving scope for reducing their operation counts.
Method
The proposed method modifies equation-solving inversion by avoiding computation of the first equation’s solution and an intermediate matrix.
Results
The proposed algorithm reduces operations by 16-17% compared with existing algorithms, while fixed-point simulations indicate good numerical accuracy.
Takeaways & Limitations
The proposed method has the lowest average norm error compared with the other methods described in the paper.
Abstract
from arXiv · showhide
In this paper we present a method for matrix inversion based on Cholesky decomposition with reduced number of operations by avoiding computation of intermediate results; further, we use fixed point simulations to compare the numerical accuracy of the method.
I. INTRODUCTION
The paper situates Cholesky- and LDL-based inversion as efficient approaches for positive-definite or symmetric matrices, then proposes reducing operations by avoiding known intermediate results.
- Cholesky- and LDL-based inversion techniques are widely used for positive-definite or symmetric matrices.
- Existing Cholesky-based inversion algorithms use equation solving or triangular matrix operations.
- The proposed algorithm reduces operations by 16-17% compared with existing algorithms by avoiding some known intermediate results.
- The paper reviews decomposition techniques, linear-system solutions, existing inversion methods, extensions to non-Hermitian matrices, and the proposed method.
II. CHOLESKY DECOMPOSITION
Cholesky decomposition factors a positive-definite Hermitian matrix into triangular components and supports matrix inversion through triangular-system solutions, with in-place implementation possible.
- II. CHOLESKY DECOMPOSITION: Cholesky decomposition factorises a positive-definite Hermitian matrix into a lower triangular matrix and its conjugate transpose.
- II. CHOLESKY DECOMPOSITION: An upper triangular representation is preferred in software because its operations are row-wise and compatible with C.
- II. CHOLESKY DECOMPOSITION: The triangular-factor elements are computed through formulas for upper triangular entries.
- II. CHOLESKY DECOMPOSITION: The factorization can be performed in-place by overwriting older a_ii values with corresponding r_ii values.
- II. CHOLESKY DECOMPOSITION: Cholesky decomposition is of order and requires operations, while inversion remains numerically stable for well conditioned matrices.
- II. CHOLESKY DECOMPOSITION: Linear systems are rewritten using the decomposition and solved through backward substitution.
A. LDL Decomposition
LDL decomposition factors a symmetric matrix into lower triangular, diagonal, and conjugate-transpose factors, avoiding square roots while matching Cholesky complexity when efficiently implemented.
- A. LDL Decomposition: LDL decomposition factorises a symmetric matrix into a lower triangular matrix, a diagonal matrix, and the lower triangular matrix’s conjugate transpose.
- A. LDL Decomposition: The decomposition eliminates the need for square-root operations.
- A. LDL Decomposition: Its elements and diagonal entries are computed using formulas for the upper triangular portion.
- A. LDL Decomposition: When efficiently implemented, LDL decomposition has the same complexity as Cholesky decomposition.
- A. LDL Decomposition: LDL-based linear systems are rewritten and solved through backward substitution.
A. Equation Solving
Matrix inversion can be obtained by solving linear systems for columns of the identity matrix, exploiting Hermitian symmetry to reduce the required computations.
- A. Equation Solving: Each inverse column is obtained by solving a linear system whose right-hand side is the corresponding identity-matrix column.
- A. Equation Solving: The system can be solved using either Cholesky- or LDL-based methods, depending on the matrix properties.
- A. Equation Solving: For Hermitian matrices, solving only the upper or lower half is sufficient because the other half follows by complex conjugation.
- A. Equation Solving: The upper half requires two triangular matrix solutions, each with multiply operations.
- A. Equation Solving: The total multiply-operation count includes both triangular solutions and the decomposition.
B. Triangular Matrix Operations
This approach derives matrix inversion through Cholesky decomposition and triangular operations, with operation counts analyzed for equation solving, multiplication, and the total computation.
- B. Triangular Matrix Operations: Cholesky decomposition is used to obtain the matrix inverse through sequential computations involving triangular matrices.The inverse is formed after computing an intermediate matrix, followed by a further computation of the inverse.
- B. Triangular Matrix Operations: Efficient equation solving and matrix multiplication are analyzed as separate operation costs.The section also includes the Cholesky decomposition in the total operation count.
C. Non-Hermitian Matrices
For non-Hermitian matrices, Cholesky or LDL decomposition is applied after constructing an intermediate Hermitian matrix, followed by operations that recover the inverse of the original matrix.
- C. Non-Hermitian Matrices: Cholesky or LDL decomposition can handle non-Hermitian matrices by first constructing an intermediate Hermitian matrix.The construction is based on an arbitrary matrix and its associated transformation.
- C. Non-Hermitian Matrices: Recovering the original inverse requires matrix transposition and matrix multiplication operations.These operations are performed after obtaining the inverse through the decomposition.
IV. PROPOSED METHOD
The proposed method modifies equation solving for Cholesky and LDL decomposition by avoiding an intermediate solution and solving only the needed upper-triangular elements before backward substitution.
- IV. PROPOSED METHOD: The proposed method avoids computing the first equation’s solution used in the existing equation-solving approach.This modification is applied to matrix inversion based on both Cholesky and LDL decomposition.
- A. Using Cholesky Decomposition: In the Cholesky variant, triangular inverse properties are used to construct the required matrix without explicitly computing the full intermediate matrix.Backward substitution then solves for the remaining quantities.
- A. Using Cholesky Decomposition: The method solves only for upper-triangle elements rather than computing all elements of the intermediate solution.The omitted lower-triangle computation is replaced by backward substitution.
- IV. PROPOSED METHOD: The proposed operation counts include multiplication operations for matrix inversion together with Cholesky or LDL decomposition.Separate totals are reported for the Cholesky and LDL variants.
- B. Using LDL Decomposition: The LDL variant constructs the corresponding intermediate matrix and uses backward substitution instead of computing the omitted intermediate result.The resulting operation count includes LDL decomposition and matrix inversion.
C. Numerical Accuracy
Fixed-point simulations report good numerical accuracy for the proposed method, which has the lowest average norm error among the methods compared.
- C. Numerical Accuracy: The proposed method has the lowest average norm error compared with the other methods evaluated in the fixed-point simulations.The simulations are used to assess numerical accuracy.
V. CONCLUSION
The proposed Cholesky-based inversion method reduces operations by avoiding intermediate computations, while fixed-point simulations evaluate numerical accuracy.
- The proposed method reduces the number of operations by avoiding computation of intermediate results.
- Table 1 summarizes the number of operations for the methods described in the paper.
- Figure 1 reports the fixed-point simulation results for the methods described in the paper.