Determining the equivalence of two sets of simultaneous linear algebraic equations
First Claim
1. A computer implemented method in a simulation of a physical system, wherein the system is described by a first set of n simultaneous linear algebraic equations and is simulated by a second system described by a second set of n simultaneous linear algebraic equations, each of said equations being of a form:
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ei1x1+ei2x2+ei3x3+ . . . +einxn=bi wherein xj are n unknowns, eij are n coefficients for each of the n equations of each set, and bi are n quantities, said coefficients and quantities being known algebraic expressions, said method comprising the steps of;
a) providing respective reduced sets, wherein step a) includes the following substeps for each of the equations of each of the sets;
a1) arranging variables in the coefficients eij and the quantities bi as multiplied instances of the variables;
a2) arranging expressions resulting from substep a1) into a form <
unitaryoperation>
<
operand>
<
operator>
<
operand>
. . . <
operator>
<
operand>
, wherein the unitary operation is either + or −
, and each operator is one of;
+, −
, or *, and wherein the substep a2) includes inserting the unitary operation in front of an expression if the expression does not already commence with the unitary operator;
a3) eliminating terms resulting from substep a2);
a4) substituting, in expressions resulting from substep a3), all + operators with a string +1* and all −
operators with a string −
1*;
a5) converting numerical terms resulting from substep a4) into an exponential format; and
a6) sorting operands of terms resulting from substep a5); and
b) combining and rearranging terms resulting from substeps a1) through a6) to eliminate said unknowns from each of said sets of simultaneous linear algebraic equations and to provide for each set, n equations in a form;
(lii)kxi=(ri)k wherein lii and ri are algebraic expressions, i={1 through n}, k=1 indicates the first one of said sets and k=2 indicates the second one of said sets; and
c) comparing, using a processor, 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 c) to determine if the products match without determining numerical values for the unknowns and without performing a matrix inversion.
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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.
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Citations
20 Claims
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1. A computer implemented method in a simulation of a physical system, wherein the system is described by a first set of n simultaneous linear algebraic equations and is simulated by a second system described by a second set of n simultaneous linear algebraic equations, each of said equations being of a form:
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ei1x1+ei2x2+ei3x3+ . . . +einxn=biwherein xj are n unknowns, eij are n coefficients for each of the n equations of each set, and bi are n quantities, said coefficients and quantities being known algebraic expressions, said method comprising the steps of; a) providing respective reduced sets, wherein step a) includes the following substeps for each of the equations of each of the sets; a1) arranging variables in the coefficients eij and the quantities bi as multiplied instances of the variables; a2) arranging expressions resulting from substep a1) into a form <
unitaryoperation>
<
operand>
<
operator>
<
operand>
. . . <
operator>
<
operand>
, wherein the unitary operation is either + or −
, and each operator is one of;
+, −
, or *, and wherein the substep a2) includes inserting the unitary operation in front of an expression if the expression does not already commence with the unitary operator;a3) eliminating terms resulting from substep a2); a4) substituting, in expressions resulting from substep a3), all + operators with a string +1* and all −
operators with a string −
1*;a5) converting numerical terms resulting from substep a4) into an exponential format; and a6) sorting operands of terms resulting from substep a5); and b) combining and rearranging terms resulting from substeps a1) through a6) to eliminate said unknowns from each of said sets of simultaneous linear algebraic equations and to provide for each set, n equations in a form;
(lii)kxi=(ri)kwherein lii and ri are algebraic expressions, i={1 through n}, k=1 indicates the first one of said sets and k=2 indicates the second one of said sets; and c) comparing, using a processor, 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 c) to determine if the products match without determining numerical values for the unknowns and without performing a matrix inversion. - View Dependent Claims (2, 3, 4, 5, 6, 7)
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8. An apparatus comprising:
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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 n simultaneous linear algebraic equations and is simulated by a second system described by a second set of n simultaneous linear algebraic equations, each of said equations being of a form;
ei1x1+ei2x2+ei3x3+ . . . +einxn=biwherein xj are n unknowns, eij are n coefficients for each of the n equations of each set, and bi are n quantities, said coefficients and quantities being known algebraic expressions, the program code comprising; instructions for performing a step a) of providing respective reduced sets, 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 <
unitaryoperation>
<
operand>
<
operator>
<
operand>
. . . <
operator>
<
operand>
, wherein the unitary operation is either + or −
, and each operator is one of;
+, −
, or *, and wherein the substep a2) includes inserting the unitary operation in front of an expression if the expression does not already commence with the unitary operator;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 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; and instructions for performing a substep a6) of sorting operands of terms resulting from substep a5); and instructions for performing a step b) of combining and rearranging terms resulting from substeps a1) through a6) to eliminate said unknowns from each of said sets of simultaneous linear algebraic equations and to provide for each set, n equations in a form;
(lii)kxi=(ri)kwherein lii and ri are algebraic expressions, i={1 through n}, and k=1 indicates the first one of said sets and k=2 indicates the second one of said sets; and instructions for performing a step c) 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 c) to determine if the products match without determining numerical values for the unknowns and without performing a matrix inversion. - View Dependent Claims (9, 10, 11, 12, 13)
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14. A computer program product for use in a simulation of a physical system, wherein the system is described by a first set of n simultaneous linear algebraic equations and is simulated by a second system described by a second set of n simultaneous linear algebraic equations, each of said equations being of a form:
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ei1x1+ei2x2+ei3x3+ . . . +einxn=biwherein xj are n unknowns, eij are n coefficients for each of the n equations of each set, and bi are n quantities, said coefficients and quantities being known algebraic expressions, the computer program product residing on a computer readable storage medium having computer readable program code, the program code comprising; instructions for performing a step a) of providing respective reduced sets, 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 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 <
unitaryoperation>
<
operand>
<
operator>
<
operand>
. . . <
operator>
<
operand>
wherein the unitary operation is either + or −
, and each operator is one of;
+, −
, or *, and wherein the substep a2) includes inserting the unitary operation in front of an expression if the expression does not already commence with the unitary operator;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; and instructions for performing a substep a6) of sorting operands of terms resulting from substep a5); and instructions for performing a step b) of combining and rearranging terms resulting from substeps a1) through a6) to eliminate said unknowns from each of said sets of simultaneous linear algebraic equations and to provide for each set, n equations in a form;
(lii)kxi=(ri)kwherein (lii) and ri are algebraic expressions, i={1 through n}, k=1 indicates the first one of said sets and k=2 indicates the second one of said sets; and instructions for performing a step c) of comparing, using a processor, 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 c) to determine if the products match without determining numerical values for the unknowns and without performing a matrix inversion. - View Dependent Claims (15, 16, 17, 18, 19, 20)
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Specification