" Yamaha However, since the correct test should be past participle, is it ok to say "Having regretted her decision, she had cancelled the order"The phrase "having regretted her decision" is non-finite. It has no tense. You are free to use any tense you like for the main clause.
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Yamaha"Having regretted her decision, she (have) cancelled the order"The participial phrase (introductory) is fine, but the subject of the main clause ("she") is singular,
Yamaha"Having regretted her decision, she (has) cancelled the order"Both ways are fine (as corrected).
Yamahais it ok to sayYes. This is also OK.
"Having regretted her decision, she had cancelled the order"