Page 10 - Linear Models for the Prediction of Animal Breeding Values

17.3. Iteration on the Mixed Model Equations 271
17.3.1. Jacobi iteration 272
17.3.2. Gauss–Seidel iteration 275
17.4. Iterating on the Data 276
17.4.1. Animal model without groups 278
17.4.2. Animal model with groups 282
17.4.3. Reduced animal model with maternal effects 284
17.5. Preconditioned Conjugate Gradient Algorithm 292
17.5.1. Computation strategy 293
17.5.2. Numerical application 294
Appendix A: Introduction to Matrix Algebra 299
A.1 Matrix: A Definition 299
A.2 Special Matrices 300
A.2.1 Square matrix 300
A.2.2 Diagonal matrix 300
A.2.3 Triangular matrix 300
A.2.4 Symmetric matrix 301
A.3 Basic Matrix Operations 301
A.3.1 Transpose of a matrix 301
A.3.2 Matrix addition and subtraction 301
A.3.3 Matrix multiplication 302
A.3.4 Direct product of matrices 302
A.3.5 Matrix inversion 303
A.3.6 Rank of a matrix 304
A.3.7 Generalized inverses 305
A.3.8 Eigenvalues and eigenvectors 305

Appendix B: Fast Algorithms for Calculating
Inbreeding Based on the L Matrix 306
B.1 Meuwissen and Luo Algorithm 306
B.1.1 Illustration of the algorithm 307
B.2 Modified Meuwissen and Luo Algorithm 308
B.2.1 Illustration of the algorithm 309

Appendix C 311
C.1 Outline of the Derivation of the Best
Linear Unbiased Prediction (BLUP) 311
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C.2 Proof that b and â from MME are the GLS
of b and BLUP of a, Respectively 312
C.3 Deriving the Equation for Progeny Contribution (PC) 313

Appendix D: Methods for Obtaining Approximate Reliability
for Genetic Evaluations 314
D.1 Computing Approximate Reliabilities for an Animal Model 314
D.2 Computing Approximate Reliabilities for Random
Regression Models 316
D.2.1 Determine value of observation for an animal 316

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