Determining the equivalence of two sets of simultaneous linear algebraic equations
3 Assignments
0 Petitions
Accused Products
Abstract
A computer implemented method (200) is described for determining the equivalence of two sets of simultaneous linear algebraic equations. Each of said equations is of a form:
ei1x1+ei2x2+ei3x3+ . . . +eiixn=bi
wherein xj are unknowns, eij are coefficients and bi are quantities, and defining the relationship between the unknowns within the set. The coefficients and quantities are known algebraic expressions. The unknowns are iteratively eliminated (250 to 280) from each of the sets of simultaneous linear algebraic equations until each of said equations are in the form:
(lii)kxi=(ri)k
wherein lii and ri are algebraic expressions, and k={1;2} indicate one of said sets that said equation is derived from. The products (lii)1*(ri)2 and (lii)2*(ri)1 are compared (300) for each of the unknowns. Only if the products match (310) for all the unknowns are the two sets of simultaneous linear algebraic equations equivalent (312). An apparatus (100) for performing the above method (200) is also provided.
-
Citations
34 Claims
-
1-14. -14. (canceled)
- 15. A computer implemented method in a simulation of a physical system, wherein the system is described by a first set of simultaneous linear algebraic equations and is simulated by a second system described by a second set of simultaneous linear algebraic equations, each of said equations being of a form:
-
22. An apparatus comprising:
-
a processor;
a storage device connected to the processor, wherein the storage device has computer readable program code for controlling the processor, and wherein the processor is operative with the program code for simulating a physical system, wherein the system is described by a first set of simultaneous linear algebraic equations and is simulated by a second system described by a second set of simultaneous linear algebraic equations, each of said equations being of a form;
ei1x1+ei2x2+ei3x3+ . . . +einxn=biwherein xj are unknowns, eij are coefficients, and bi are quantities, said coefficients and quantities being known algebraic expressions, the program code comprising;
instructions for performing a step a) of eliminating said unknowns from each of said sets of simultaneous linear algebraic equations, wherein the instructions include;
instructions for performing a substep a1) of arranging variables in the coefficients eij and the quantities bi in a form having multiplied instances of the variables;
instructions for performing a substep a2) of arranging expressions resulting from substep a1) into a form having a string of operations on operands without any division operations;
instructions for performing a substep a3) of eliminating terms resulting from substep a2);
instructions for performing a substep a4) of substituting, in expressions resulting from substep a3), all + operators with a string +1* and all −
operators with a string −
1*;
instructions for performing a substep a5) of converting numerical terms resulting from substep a4) into an exponential format;
instructions for performing a substep a6) of sorting operands of terms resulting from substep a5); and
instructions for performing a substep a7) of combining and rearranging terms resulting from substep a6) to reduce such an equation to a form;
(lii)kxi=(ri)kwherein lii and ri are algebraic expressions, and k={1;
2} indicate one of said sets that said equation is derived from; and
instructions for performing a step b) of comparing, for each of said unknowns, a first product (lii)1*(ri)2 and a second product (lii)2*(ri)1, wherein the first product is an algebraic expression and the second product is an algebraic expression, and wherein if said products match for all said unknowns said second set of simultaneous linear algebraic equations is equivalent to the first set of simultaneous linear algebraic equations and thereby is determined to be a proper representation of the physical system, wherein the eliminating said unknowns in step a) enables the comparing in step b) to determine if the products match without determining numerical values for the unknowns and without performing a matrix inversion. - View Dependent Claims (23, 24, 25, 26, 27)
-
-
28. A computer program product for use in a simulation of a physical system, wherein the system is described by a first set of simultaneous linear algebraic equations and is simulated by a second system described by a second set of simultaneous linear algebraic equations, each of said equations being of a form:
-
ei1x1+ei2x2+ei3x3+ . . . +einxn=biwherein xj are unknowns, eij are coefficients, and bi are quantities, said coefficients and quantities being known algebraic expressions, the computer program product residing on a computer usable medium having computer readable program code, the program code comprising;
instructions for performing a step a) of eliminating said unknowns from each of said sets of simultaneous linear algebraic equations, wherein the instructions include;
instructions for performing a substep a1) of arranging variables in the coefficients eij and the quantities bi in a form having multiplied instances of the variables;
instructions for performing a substep a2) of arranging expressions resulting from substep a1) into a form having a string of operations on operands without any division operations;
instructions for performing a substep a3) of eliminating terms resulting from substep a2);
instructions for performing a substep a4) of substituting, in expressions resulting from substep a3), all + operators with a string +1* and all −
operators with a string −
1*;
instructions for performing a substep a5) of converting numerical terms resulting from substep a4) into an exponential format;
instructions for performing a substep a6) of sorting operands of terms resulting from substep a5); and
instructions for performing a substep a7) of combining and rearranging terms resulting from substep a6) to reduce such an equation to a form;
(lii)kxi=(ri)kwherein lii and ri are algebraic expressions, and k={1;
2} indicate one of said sets that said equation is derived from; and
instructions for performing a step b) of comparing, for each of said unknowns, a first product (lii)1*(ri)2 and a second product (lii)2*(ri)1, wherein the first product is an algebraic expression and the second product is an algebraic expression, and wherein if said products match for all said unknowns said second set of simultaneous linear algebraic equations is equivalent to the first set of simultaneous linear algebraic equations and thereby is determined to be a proper representation of the physical system, wherein the eliminating said unknowns in step a) enables the comparing in step b) to determine if the products match without determining numerical values for the unknowns and without performing a matrix inversion. - View Dependent Claims (29, 30, 31, 32, 33, 34)
-
Specification