Does God Ever Have Regrets?

This morning in my Bible reading I came across a passage that has puzzled me in the past. In 1 Sam 15, King Saul has finally crossed the line. God is going to replace him with another king. And this is what 15:10-11 says:

The word of the LORD came to Samuel: (11) "I regret that I have made Saul king, for he has turned back from following me and has not performed my commandments." And Samuel was angry, and he cried to the LORD all night.

I also read to the end of the chapter where it says:

And Samuel did not see Saul again until the day of his death, but Samuel grieved over Saul. And the LORD regretted that he had made Saul king over Israel. (v 35)

The word that is used for "regret" could also be translated "repent" or "changed his mind."

How can the all-wise God, who is sovereign over everything regret, repent, or change his mind?

When Balaam spoke the words of God to Balek he said this:

God is not man, that he should lie, or a son of man, that he should change his mind [repent]. Has he said, and will he not do it? Or has he spoken, and will he not fulfill it? (Numbers 23:19)

In fact, back in 1 Sam 15:29 it says about God ("The Glory of Israel"):

And also the Glory of Israel will not lie or have regret, for he is not a man, that he should have regret."

Here are two helpful words from a old timer and one still on the scene:

John Calvin in His Institutes on "God's Repentance"
John Piper -- "God Does Not Repent Like A Man"