# What is a "secular equation"?

Golub mentioned "secular equation" in his Matrix Computation and a slide. However I still don't get its definition. How is a secular equation defined? Thanks!

A secular equation is another name for the characteristic equation.

http://en.wikipedia.org/wiki/Characteristic_polynomial#Secular_equation

The reason it is called "secular" is because it was first used in calculations relating the planetary motion:

http://en.wikipedia.org/wiki/Secular_phenomena

http://en.wikipedia.org/wiki/Secular_variation

The etymological root of "secular" is "saeculum," meaning "of an age."

• Why it is called secular equation? Is there also a sacred equation? Mar 20, 2013 at 12:56
• @Popopo See my edits above. Mar 20, 2013 at 13:04
• @steveO: secular here means one direction of change with time, as opposed to periodic. If you have $f(t)=at+b \sin t$, the $at$ is the secular part and the $b \sin t$ is the periodic part. Mar 20, 2013 at 13:22
• It's helpful. Thank you. Mar 20, 2013 at 13:29
• An attempt to clarify "secular" vs. "sacred": As far as I know, "saeculum" can mean either "age" or "world". The former meaning leads to secular equations, the latter leads to non-sacred things. Mar 20, 2013 at 13:30