Efficient methods for genomic prediction were developed to improve reliability of estimated breeding values and estimate thousands of marker effects simultaneously. Algorithms were tested with simulated data for 2,967 bulls and 50,000 markers across 30 chromosomes. Linear and nonlinear methods were used to predict breeding values. Linear methods included selection index and mixed model equations, while nonlinear methods used iteration and regression on marker deviations. A blend of first- and second-order Jacobi iteration with relaxation factors converged well for allele frequencies and effects. Reliability of predicted net merit for young bulls was 63% compared to 32% using traditional relationship matrices. Nonlinear predictions increased reliability to 66%. Computing times increased linearly with the number of genotypes. Allele frequency estimation required 2 processor days, while genomic predictions took less than 1 day per trait. Genotyping information was equivalent to about 20 daughters with phenotypic records. Reliability gains may vary due to factors like linkage disequilibrium not accounted for in the simulation. Genomic relationships were computed using three methods: a formula based on marker deviations, a method weighting markers by variance, and a regression-based method. Genomic inbreeding coefficients were calculated using these methods, with the first method being more precise when using base population allele frequencies. The second method was less precise, while the third method was more accurate when adjusting for mean homozygosity. Genomic predictions were compared with traditional methods, showing higher reliability for young bulls. Nonlinear predictions slightly outperformed linear predictions in reliability. Reliability for older bulls was higher, with gains in reliability being small due to already high reliability levels. Genomic predictions were computed using linear and nonlinear models, with nonlinear models showing slightly better correlations with true breeding values. Reliability for young bulls was 66% using nonlinear predictions, compared to 63% using linear predictions. Reliability for older bulls was 91%, compared to 90% using traditional methods. Genomic predictions were more accurate than traditional methods, with reliability gains being small for older bulls due to high baseline reliability. Reliability gains were converted to daughter equivalents, showing that genotyping information was equivalent to including records from about 20 additional daughters. New methods are needed to explain genomic predictions due to limited understanding among scientists and breeders. Genomic predictions were applied to actual data, showing higher reliability for young animals and bulls with daughters. The study concluded that genomic selection improves genetic improvement and reduces progeny testing costs, with computational methods being essential for processing large genomic datasets.Efficient methods for genomic prediction were developed to improve reliability of estimated breeding values and estimate thousands of marker effects simultaneously. Algorithms were tested with simulated data for 2,967 bulls and 50,000 markers across 30 chromosomes. Linear and nonlinear methods were used to predict breeding values. Linear methods included selection index and mixed model equations, while nonlinear methods used iteration and regression on marker deviations. A blend of first- and second-order Jacobi iteration with relaxation factors converged well for allele frequencies and effects. Reliability of predicted net merit for young bulls was 63% compared to 32% using traditional relationship matrices. Nonlinear predictions increased reliability to 66%. Computing times increased linearly with the number of genotypes. Allele frequency estimation required 2 processor days, while genomic predictions took less than 1 day per trait. Genotyping information was equivalent to about 20 daughters with phenotypic records. Reliability gains may vary due to factors like linkage disequilibrium not accounted for in the simulation. Genomic relationships were computed using three methods: a formula based on marker deviations, a method weighting markers by variance, and a regression-based method. Genomic inbreeding coefficients were calculated using these methods, with the first method being more precise when using base population allele frequencies. The second method was less precise, while the third method was more accurate when adjusting for mean homozygosity. Genomic predictions were compared with traditional methods, showing higher reliability for young bulls. Nonlinear predictions slightly outperformed linear predictions in reliability. Reliability for older bulls was higher, with gains in reliability being small due to already high reliability levels. Genomic predictions were computed using linear and nonlinear models, with nonlinear models showing slightly better correlations with true breeding values. Reliability for young bulls was 66% using nonlinear predictions, compared to 63% using linear predictions. Reliability for older bulls was 91%, compared to 90% using traditional methods. Genomic predictions were more accurate than traditional methods, with reliability gains being small for older bulls due to high baseline reliability. Reliability gains were converted to daughter equivalents, showing that genotyping information was equivalent to including records from about 20 additional daughters. New methods are needed to explain genomic predictions due to limited understanding among scientists and breeders. Genomic predictions were applied to actual data, showing higher reliability for young animals and bulls with daughters. The study concluded that genomic selection improves genetic improvement and reduces progeny testing costs, with computational methods being essential for processing large genomic datasets.