This article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
First, it ignores work by Gauss predecessors, including Newton. Second, it ignores the real world drivers (astronomy, business, and military issues) behind the effort to establish quicker tabulation methods. -- 70.112.146.196 17:30, 8 July 2006 (UTC)
In iterative methods, there may be cases where the cost of moving from one point to the next varies with location (e.g., it might cost more to move from (1,0,0,0) to (4,3,2,1) than to (2,0,0,0)); there also may be cases where the cost of evaluation is location-dependent (e.g., evaluating f(1000,1000) may cost less than evaluating f(42,37)). In such cases, the optimal "next guess" would be different than if these constraints did not exist. Is there literature on this sort of thing? —Ben FrantzDale 04:53, 27 April 2007 (UTC)
Standard division algorithms – restoring and non-restoring procedures - iterative produce one digit of the final quotient per iteration. The division methods are all based on the form X = b / A, where • X = Quotient • b = Dividend • A = Divisor Division methods are all based on a standard recurrence equation:
Wk+1 =2Wk - Aqk,
where: • Wk = the partial remainder of the division, • qk = the digit of the quotient X in k-th position, • A = the divisor.
Both restoring and non-restoring divisions are based on the analysis of the sign of the partial remainder Wk. If Wk >= 0, then the divisor A is subtracted from the partial remainder Wk , if Wk < 0 then the divisor A is added to the partial remainder Wk . Both methodes doesn’t take into account the value of the partial remainder Wk , only its sign.
Taking into account the value of the most significant digit of the partial remainder Wk , can accelerate the division process. If the most significant digit of the Wk is “1”, then we choose according to the sign of Wk - addition or subtraction the divisor A to the Wk and write -1 or +1 to the correspondingly position of the quotient X as in non-restoring division procedure.
But if the most significant digit of the Wk is “0”, then we skip this iteration and write “0” to the correspondingly position of the quotient X . So we get the quotient in the ternary system with the set of digits: {+1, 0, -1}, which is converted then very easy into traditional binary system.
Evidently this approach can be generalized to the vectors and matrices. For example if A - m-th order matrix, b – the right part of the linear system of equations, X – the vector of the unknowns, we have: X = b A-1 , or: AX - b = 0 For the k-th iteration: AXk - b = Wk, where:
Wk - value of residual vector for k-th iteration; Xk - value of result vector of unknowns on k-th iteration.
vladimir@baykov.de — Preceding unsigned comment added by Vladimir Baykov ( talk • contribs) 14:52, 12 March 2011 (UTC)
As per numerical analysis, "iterative methods form successive approximations that converge to the exact solution". fgnievinski ( talk) 02:56, 21 December 2023 (UTC) fgnievinski ( talk) 02:56, 21 December 2023 (UTC)
This article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
First, it ignores work by Gauss predecessors, including Newton. Second, it ignores the real world drivers (astronomy, business, and military issues) behind the effort to establish quicker tabulation methods. -- 70.112.146.196 17:30, 8 July 2006 (UTC)
In iterative methods, there may be cases where the cost of moving from one point to the next varies with location (e.g., it might cost more to move from (1,0,0,0) to (4,3,2,1) than to (2,0,0,0)); there also may be cases where the cost of evaluation is location-dependent (e.g., evaluating f(1000,1000) may cost less than evaluating f(42,37)). In such cases, the optimal "next guess" would be different than if these constraints did not exist. Is there literature on this sort of thing? —Ben FrantzDale 04:53, 27 April 2007 (UTC)
Standard division algorithms – restoring and non-restoring procedures - iterative produce one digit of the final quotient per iteration. The division methods are all based on the form X = b / A, where • X = Quotient • b = Dividend • A = Divisor Division methods are all based on a standard recurrence equation:
Wk+1 =2Wk - Aqk,
where: • Wk = the partial remainder of the division, • qk = the digit of the quotient X in k-th position, • A = the divisor.
Both restoring and non-restoring divisions are based on the analysis of the sign of the partial remainder Wk. If Wk >= 0, then the divisor A is subtracted from the partial remainder Wk , if Wk < 0 then the divisor A is added to the partial remainder Wk . Both methodes doesn’t take into account the value of the partial remainder Wk , only its sign.
Taking into account the value of the most significant digit of the partial remainder Wk , can accelerate the division process. If the most significant digit of the Wk is “1”, then we choose according to the sign of Wk - addition or subtraction the divisor A to the Wk and write -1 or +1 to the correspondingly position of the quotient X as in non-restoring division procedure.
But if the most significant digit of the Wk is “0”, then we skip this iteration and write “0” to the correspondingly position of the quotient X . So we get the quotient in the ternary system with the set of digits: {+1, 0, -1}, which is converted then very easy into traditional binary system.
Evidently this approach can be generalized to the vectors and matrices. For example if A - m-th order matrix, b – the right part of the linear system of equations, X – the vector of the unknowns, we have: X = b A-1 , or: AX - b = 0 For the k-th iteration: AXk - b = Wk, where:
Wk - value of residual vector for k-th iteration; Xk - value of result vector of unknowns on k-th iteration.
vladimir@baykov.de — Preceding unsigned comment added by Vladimir Baykov ( talk • contribs) 14:52, 12 March 2011 (UTC)
As per numerical analysis, "iterative methods form successive approximations that converge to the exact solution". fgnievinski ( talk) 02:56, 21 December 2023 (UTC) fgnievinski ( talk) 02:56, 21 December 2023 (UTC)