Methods and apparatus to solve for a minimum variance solution for a constrained weight problem in a scalable and memory efficient manner are disclosed. An example method of reducing memory usage of a computing device disclosed herein comprises storing a constraint matrix and a population matrix into a memory, the population matrix representing a population related to data of the constraint matrix, determining, via a processor, a linear system based on a set of constraints including the constraint matrix and the population matrix, determining, via a processor, an inverse of a matrix of the linear system, the determination of the inverse including breaking down the matrix into blocks, the dimensions of the blocks being no larger than dimensions of the constraint matrix, and outputting a set of weights based on the inverse of the linear system, the set of weights satisfying the set of constraints.
Please help me write a proper abstract based on the patent claims. CLAIM: 1. A method for determining a plurality of weights for a plurality of panelist representative of a population, the method comprising: storing a constraint matrix and a population matrix into a memory, the constraint matrix representing first demographics associated with a plurality of panelists in a population, the population matrix representing second demographics of the population; determining, via a processor, a linear system based on a set of constraints including the constraint matrix and the population matrix; determining, via a processor, an inverse of a first matrix of the linear system, the determination of the inverse including breaking down the first matrix into blocks, the dimensions of the blocks being no larger than dimensions of the constraint matrix; and outputting a resultant set of weights based on the inverse of the linear system, the resultant set of weights satisfying the set of constraints. 2. The method of claim 1, wherein the linear system is a predictor linear system, and further including: determining a first solution to the predictor linear system; determining a corrector linear system based on the first solution; solving the corrector linear system based on the set of constraints to determine an adjusted solution; and adjusting an initial set of weights based on an inverse of a second matrix of the corrector linear system, wherein the resultant set of weights is further based on the adjusted initial set of weights. 3. The method of claim 2, wherein a second solution to the corrector linear system is calculated multiple times until the adjusted initial set of weight converges to the resultant set of weights, the multiple calculations based on prior solutions to the corrector linear system. 4. The method of claim 1, further including applying the resultant set of weights to data associated with the panelists. 5. The method of claim 4, wherein the applying of the resultant set of weights includes: receiving a media exposure event from a panelist from the plurality of panelists with a weight from the set of weights; and crediting demographics of the panelist for the media exposure event based on the weight. 6. The method of claim 1, wherein the resultant set of weights is an optimal solution to an equality constraint, the equality constraint being an equation including the constraint matrix, the population matrix, and the set of weights. 7. The method of claim 6, wherein: the set of constraints further includes at least one of a non-negative constraint, a bound constraint, and the equality constraint; and the optimal solution is a non-negative solution with minimum variance. 8. The method of claim 1, wherein the inverse of the first matrix of the linear system is generated by: separating components of the first matrix into a first set of blocks; inverting the first matrix using blockwise inversion, the inverse of the first matrix being a second matrix; separating a first element of the second matrix into a second set of blocks; inverting the second matrix using blockwise inversion, the inverse of the second matrix being a third matrix; determining a second element of the third matrix whose dimensions are larger than the constraint matrix; reducing the dimensions of the second element to a size less than or equal to the dimensions of the constraint matrix, the reduced second element being a third element; replacing the second element with the third element in the second matrix; and replacing the first element with the second matrix. 9. The method of claim 8, wherein the dimensions of the second element are reduced using Sherman-Morrison formula. 10. The method of claim 9, wherein the memory is conserved by eliminating the need to store the linear system. 11. An apparatus to determine a plurality of weights for a plurality of panelist representative of a population, the apparatus comprising: memory to store a constraint matrix and a population matrix, the constraint matrix representing first demographics associated with a plurality of panelists in a population, the population matrix representing second demographics of the population; and a weight determiner to: determine a linear system based on a set of constraints including the constraint matrix and the population matrix; determine an inverse of a first matrix of the linear system, the determination of the inverse including breaking down the first matrix into blocks, the dimensions of the blocks being no larger than dimensions of the constraint matrix; and output a resultant set of weights based on the inverse of the linear system, the resultant set of weights satisfying the set of constraints. 12. The apparatus of claim 11, wherein the linear system is a predictor linear system, and the weight determiner is to: determine a first solution to the predictor linear system; determine a corrector linear system based on the first solution; solve the corrector linear system based on the set of constraints to determine an adjusted solution; and adjust an initial set of weights based on an inverse of a second matrix of the corrector linear system, wherein the resultant set of weights is further based on the adjusted initial set of weights. 13. The apparatus of claim 12, wherein a second solution to the corrector linear system is calculated multiple times until the adjusted initial set of weight converges to the resultant set of weights, the multiple calculations based on prior solutions to the corrector linear system. 14. The apparatus of claim 11, wherein the weight determiner is to apply the resultant set of weights to data associated with the panelists. 15. The apparatus of claim 14, wherein the weight determiner is to apply the resultant set of weights by: receiving a media exposure event from a panelist from the plurality of panelists with a weight from the set of weights; and crediting demographics of the panelist for the media exposure event based on the weight. 16. The apparatus of claim 11, wherein the resultant set of weights is an optimal solution to an equality constraint, the equality constraint being an equation including the constraint matrix, the population matrix, and the set of weights. 17. The apparatus of claim 16, wherein: the set of constraints further includes at least one of a non-negative constraint, a bound constraint, and the equality constraint; and the optimal solution is a non-negative solution minimum variance. 18. The apparatus of claim 11, wherein the weight determiner is to determine the inverse of the matrix of the linear system by: separating components of the first matrix into a first set of blocks; inverting the first matrix using blockwise inversion, the inverse of the first matrix being a second matrix; separating a first element of the second matrix into a second set of blocks; inverting the second matrix using blockwise inversion, the inverse of the second matrix being a third matrix; determining a second element of the third matrix whose dimensions are larger than the constraint matrix; reducing the dimensions of the second element to a size less than or equal to the dimensions of the constraint matrix, the reduced second element being a third element; replacing the second element with the third element in the second matrix; and replacing the first element with the second matrix. 19. (canceled) 20. (canceled) 21. A computer readable medium comprising instructions that, when executed cause a machine to at least: store a constraint matrix and a population matrix into memory, the constraint matrix representing first demographics associated with a plurality of panelists in a population, the population matrix representing second demographics of the population; determine a linear system based on a set of constraints including the constraint matrix and the population matrix; determine an inverse of a first matrix of the linear system, the determination of the inverse including breaking down the first matrix into block, the dimensions of the blocks being no larger than dimensions of the constraint matrix; and output a resultant set of weights based on the inverse of the linear system, the resultant set of weights satisfying the set of constraints. 22. The computer readable medium of claim 21, wherein the linear system is a predictor linear system, and further including instructions to cause the machine to: determine a first solution to the predictor linear system; determine a corrector linear system based on the first solution; solve the corrector linear system based on the set of constraints to determine an adjusted solution; and adjust an initial set of weights based on an inverse of a second matrix of the corrector linear system, wherein the resultant set of weights is further based on the adjusted initial set of weights. 23. The computer readable medium of claim 22, wherein the instructions cause a machine to calculate a second solution to the corrector linear system multiple times until the adjusted initial set of weight converges to the resultant set of weights, the multiple calculations based on prior solutions to the corrector linear system. 24. (canceled) 25. (canceled) 26. The computer readable medium of claim 21, wherein the resultant set of weights is an optimal solution to an equality constraint, the equality constraint being an equation including the constraint matrix, the population matrix, and the set of weights. 27. The computer readable medium of claim 26, wherein: the set of constraints further includes at least one of a non-negative constraint, a bound constraint, and the equality constraint; and the optimal solution is a non-negative solution with minimum variance. 28. The computer readable medium of claim 21, wherein the instructions cause the machine to determine the inverse of the first matrix of the linear system by: separating components of the first matrix into a first set of blocks; inverting the first matrix using blockwise inversion, the inverse of the first matrix being a second matrix; separating a first element of the second matrix into a second set of blocks; inverting the second matrix using blockwise inversion, the inverse of the second matrix being a third matrix; determining a second element of the third matrix whose dimensions are larger than the constraint matrix; reducing the dimensions of the second element to a size less than or equal to the dimensions of the constraint matrix, the reduced second element being a third element; replacing the second element with the third element in the second matrix; and replacing the first element with the second matrix. 29. (canceled) 30. (canceled) 31. (canceled)