Determining ill conditioning in square linear system of equations

• US 8,712,738 B2
• Filed: 04/30/2010
• Issued: 04/29/2014
• Est. Priority Date: 04/30/2010
• Status: Active Grant

• Current Assignee: International Business Machines Corporation
• Original Assignee: International Business Machines Corporation
• Inventors: Danna, Emilie Jeanne Anne, Klotz, Edward Seymour
• Primary Examiner(s): Rivas, Omar Fernandez
• Assistant Examiner(s): GEBRESILASSIE, KIBROM K
• Application Number:
  US12/772,062
• Publication Number:
  US 20110270589A1
• Time in Patent Office: 1,460 Days
• Field of Search: 703/2, 705/14.52
• US Class Current: 703/2
• CPC Class Codes: G06F 17/12 Simultaneous equations , e.g. systems of linear equations

Abstract

Determining ill conditioning in square linear systems of equations is disclosed. Specifically, a method, system and computer program product for determining ill conditioning in square linear systems of equations is disclosed, the method including sampling condition numbers in the square linear systems of equations, and classifying the condition numbers sampled from the square linear systems of equations.

Claims

1. A method for determining ill conditioning in square linear systems of equations, the method comprising:

sampling, by a processor, a plurality of condition numbers in the square linear systems of equations;

classifying the condition numbers sampled from the square linear systems of equations in different categories;

generating statistics information on the square linear systems of equations for each of the different categories of the classifying; and

displaying the statistics information and the different categories according to classification of each of the plurality of condition numbers on a display and, depending on a value of the statistics information, displaying notification that the square linear systems of equations are ill conditioned.

2. The method according to claim 1, further comprising:

assessing a likelihood of ill conditioning according to the generated statistics information.

3. The method according to claim 1, wherein the classifying the condition numbers comprises:

obtaining a machine precision value of a computer;

obtaining a tolerance value of an application operating on the computer; and

calculating a threshold value which distinguishes large and small condition numbers according to the machine precision value of the computer and the tolerance value of an application.

4. The method according to claim 1, wherein the different categories are determined according to a level of ill conditioning.

5. The method according to claim 4, wherein the level of ill conditioning comprises one of a stable category, a suspicious category, an unstable category and an ill-posed category.

6. The method according to claim 3, wherein if the condition number is greater than or equal to the threshold value, the square linear system is identified as ill conditioned.

7. The method according to claim 3, wherein if the condition number is less than the threshold value, the square linear system is identified as unlikely to be ill conditioned.

8. The method according to claim 1, wherein the sampling is obtained for a predetermined amount of condition numbers.

9. A non-transitory computer readable medium storing instructions for determining ill conditioning in linear systems of equations, the instructions comprising:

sampling a plurality of condition numbers in the square linear systems of equations;

classifying the condition numbers sampled from the square linear systems of equations in different categories;

generating statistics information on the square linear systems of equations for each of the different categories of the classifying; and

displaying the statistics information and the different categories according to classification of each of the plurality of condition numbers on a display and, depending on a value of the statistics information, displaying notification that the square linear systems of equations are ill conditioned.

10. The computer readable medium according to claim 9, wherein the classifying the condition numbers comprises:

obtaining a machine precision value of a computer;

obtaining a tolerance value of an application operating on the computer; and

calculating a threshold value which distinguishes large and small condition numbers according to the machine precision value of the computer and the tolerance value of an application.

11. The computer readable medium according to claim 9, wherein the different categories are determined according to a level of ill conditioning.

12. The computer readable medium according to claim 11, wherein the level of ill conditioning comprises one of a stable category, a suspicious category, an unstable category and an ill-posed category.

13. The computer readable medium according to claim 10, wherein if the condition number is greater than or equal to the threshold value, the square linear system from which the condition number was computed is identified as ill conditioned.

14. The computer readable medium according to claim 10, wherein if the condition number is less than the threshold value, the square linear system from which the condition number was computed is identified as unlikely to be ill conditioned.

15. The computer readable medium according to claim 9, wherein the sampling is obtained for a predetermined amount of condition numbers.

16. A system for determining ill conditioning in linear systems of equations, the system comprising:

a processor with a memory,

a sampling unit which samples a plurality of condition numbers in the square linear systems of equation;

a classifying unit which classifies the condition numbers sampled from the square linear systems of equations in different categories;

a statistical analysis unit which generates statistics information on the square linear systems of equations for each of the different categories of the classifying in the classifying unit; and

a display which displays the statistics information and the different categories according to classification of each of the plurality of condition numbers and, depending on a value of the statistics information, displays notification that the squares linear systems of equations are ill conditioned.

17. The system according to claim 16, further comprising:

an assessment unit which assesses a likelihood of ill conditioning according to the generated statistics information.

18. The system according to claim 16, wherein the classifying unit classifies the condition numbers by:

obtaining a machine precision value of a computer;

obtaining a tolerance value of an application operating on the computer; and

calculating a threshold value which distinguishes large and small condition numbers according to the machine precision value of the computer and the tolerance value of an application.

19. The system according to claim 16, wherein the different categories are determined according to a level of ill conditioning.

20. A method for determining ill conditioning in a plurality of linear systems of equations, the method comprising:

sampling, by processor, a plurality of condition numbers in the plurality of the square linear systems of equations;

classifying the condition numbers sampled from the plurality of square linear systems of equations in different categories;

generating statistics information on each of the plurality of square linear systems of equations for each of the different categories of the classifying;

assessing a likelihood of ill conditioning according to the generated statistics information; and

displaying the statistics information and the different categories according to classification of each of the plurality of condition numbers on a display and, depending on a value of the statistics information, displaying notification that the square linear systems of equations are ill conditioned.

21. A method for determining ill conditioning in a square linear system of equations, the method comprising:

obtaining, by a processor, a plurality of condition numbers of the square linear system of equations;

assessing whether the square linear system is ill conditioned according to each of different categories of the condition numbers;

generating statistics information on the square linear systems of equations for the each of the different categories; and

displaying the statistics information and the different categories according to classification of each of a plurality of the condition numbers on a display and, depending on a value of the statistics information, displaying notification that the square linear systems of equations are ill conditioned.

22. The method according to claim 21, wherein the different categories comprise one of a stable category, a suspicious category, an unstable category, and an ill-posed category.