Algorithm to minimize iterative computation in a process for the analysis or design of a physical system

Claim 2

In a process for the analysis or design of a physical system, which system is represented by algebraic equations, some or all of which may be non-linear, with a minimum of iterative calculation, by solving said equations in the order selected by a computational sequence and thereby determining numerical values for variables in said algebraic equations, the improvement comprising in combination determining said computational sequence by use of a programmed computing system having stored in its internal memory a program enabling the computing machine to perform in combination the steps of: a. forming a single system structural matrix in which each row is an equation of said system and each column is a variable of said system, or vice-versa, and wherein each variable is designated in each equation as being either known or unknown, b. identifying design and/or iterative variables by algebraic operations on the system structural matrix so as to minimize the number of said iterative variables, and identifying iterative variable hierarchy levels, when present so as to minimize nested iterative calculations, c. selecting the computational sequence for solution of said equations, said sequence being selected by algebraic manipulation of the system structural matrix.

Claim 3

The process of claim 1 wherein the algebraic operations which minimize the number of iterative variables comprise in combination: a. computing the frequency of occurrence of each variable in the system structural matrix, b. eliminating variables with zero frequency of occurrence from the system structural matrix, said zero frequency variables being already known, the number of variables in said system structural matrix being thereby reduced, c. computing the number of unknown variables remaining in each equation, d. if unknown variables still remain, recomputing the frequency of occurrence of each variable, e. determining the minimum frequency of occurrence of unknown variables and then redesignating as known, equations which contain the minimum frequency variables until the number of equations redesignated as known equals the minimum frequency of occurrence minus one, f. recomputing the frequency of occurrence of each unknown variable and indicating that minimum frequency minus one iterative variables have been identified as being within the remaining unknown variables, indicating that a hierarchy level has been detected, g. redesignating as known any unknown variable which appears in only one equation after step d. and also redesignating the equation as known, provided that said equation has not been previously designated as known, h. designating as ''''design or iterative variables'''' each unknown variable which appears in only one equation after step d. said equation having been redesignated as known in accordance with step g. above, said ''''design or iterative'''' variables associated with any hierarchy level designated by a number less than or equal to the number designating the present hierarchy level.

Claim 4

The process of claim 1 wherein the algebraic manipulations for selecting the computational sequence for solution of said equations so as to minimize nested iterative calculations comprises in combination the steps of: a. selecting a set of design variables and designating them as known, b. determining the frequency of occurrence of unknown variables and, also, computing the number of unknown variables in each equation of the system structural matrix, c. redesignating as known those variables which appear by themselves in an equation and redesignating as known that equation; but if it is determined that the minimum number of unknown variables remaining in each equation is greater than one, then, d. proceeding by temporarily designating as known all iterative variables appearing in the hierarchy level under consideration, e. redesignating as known, an unknown variable with a frequency of occurrence greater than one and simultaneously redesignating as known an equation in which the only unknown variable is that unknown variable which equation also contains at least one iterative variable, f. checking to see if the number of unknown variables in each equation of the system structural matrix is equal to zero and if the number is equal to zero solving for one iterative variable in terms of the other(s), g. checking to see if the number of equations redesignated as known is equal to the number of essential iterative variables in the hierarchy level under consideration, and if the number is less than the number of essential iterative variables, proceeding in accordance with step d., above, h. redesignating as unknown those iterative variables within the iterative variable hierarchy under study, which have not been redesignated as known in accordance with step d. above, i. if all system variables have been redesignated as known, designating as the calculational sequence, the order in which the system variable and the corresponding equation have been redesignated as known, j. if unknown variables still remain, recomputing the frequency of occurrence of each variable and by computing the number of unknown variables remaining in each equation and proceeding in accordance with step c. above.

Claim 5

The process of claim 2 wherein the algebraic manipulations for selecting the computational sequence for solution of said equations comprises in combination the steps of; a. selecting a set of design variables and designating them as known, b. determining the frequency of occurrence of unknown variables and, also, computing the number of unknown variables in each equation of the system structural matrix, c. redesignating as known those variables which appear by themselves in an equation and redesignating as known that equation; but if it is determined that the number of unknown variables remaining in each equation is greater than one, then, d. proceeding by temporarily designating as known all iterative variables appearing in the hierarchy level under consideration, e. redesignating as known an unknown variable with a frequency of occurrence greater than one and simultaneously redesignating as known an equation in which the only unknown variable is that unknown variable which equation also contains at least one iterative variable, f. checking to see if the number of unknown variables in each equation of the system structural matrix is equal to zero and if the number is equal to zero solving for one iterative variable in terms of the other(s), g. checking to see if the number of equations redesignated as known is equal to the number of essential iterative variables in the hierarchy level under consideration, and if the number is less than the number of essential iterative variables, proceeding in accordance with step d., above, h. redesignating as unknown those iterative variables within the hierarchy under study, which have not been redesignated as known in accordance with step d. above, i. if all system variables have been redesignated as known, designating as the calculational sequence, the order in which the system variable and the corresponding equation have been redesignated as known, j. if unknown variables still remain, recomputing the frequency of occurrence of each variable and by computing the number of unknown variables remaining in each equation and proceeding in accordance with step c. above.

Claim 6

The process of claim 1 wherein the algebraic operations which minimize the number of iterative variables comprise in combination: a. computing the frequency of occurrence of each variable in the system structural matrix, b. eliminating variables with zero frequency of occurrence from the system structural matrix, said zero frequency variables being already known, the number of variables in said system structural matrix being thereby reduced, c. computing the number of unknown variables remaining in each equation, d. if unknown variables still remain, recomputing the frequency of occurrence of each variable, e. determining the minimum frequency of occurrence of unknown variables and then redesignating as known, equations which contain the minimum frequency variables until the number of equations redesignated as known equals the minimum frequency of occurrence minus one, f. recomputing the frequency of occurrence of each unknown variable and indicating that minimum frequency minus one iterative variables have been identified as being within the remaining unknown variables, indicating that a hierarchy level has been detected, g. redesignating as known any unknown variable which appears in only one equation after step d. and also redesignating the equation as known, provided that saiD equation has not been previously designated as known, h. designating as ''''design or iterative variables'''' each unknown variable whose variable frequency becomes zero after step d. and whose equations have been redesignated as known in accordance with step g. above, said ''''design or iterative'''' variables associated with any hierarchy level designated by a number less than or equal to the number designating the present hierarchy level.

Claim 7

A process according to claim 1 wherein said structural matrix is descriptive of a system in the cost analysis art.

Claim 8

A process according to claim 1 wherein said structural matrix is descriptive of a system in the plant design art.

Claim 9

A process according to claim 1 wherein said structural matrix is descriptive of a system in the electronics art.

Claim 10

A process according to claim 1 wherein said structural matrix is descriptive of a system in the distribution analysis art.