SAN FRANCISCO » The two biggest — and bitterest — rivals in the smartphone market will have to endure another bruising trial after a federal judge ruled that jurors miscalculated nearly half the $1 billion in damages it found Samsung Electronics owed Apple Inc. for patent infringement.

U.S. District Judge Lucy Koh wiped out $450 million from the verdict and ordered a new trial to reconsider damages related to 14 Samsung products including some products in its hot-selling Galaxy lineup jurors in August found were using Apple’s technology without permission. Koh said jurors in three-week trial had not properly followed her instruction in calculating some of the damages.

She also concluded that mistakes had been made in determining when Apple had first notified Samsung about the alleged violations of patents for its trend-setting iPhone and IPad.

“We are pleased that the court decided to strike $450,514,650 from the jury’s award,” Samsung spokeswoman Lauren Restuccia said.

Koh didn’t toss out the jurors underlying finding that two dozen Samsung products infringed patents Appleused to develop its iPad and iPhone products. The new jury will be tasked with only determining what Samsung owes Apple.

Apple declined to comment on the Koh’s ruling, which still did leave Samsung with a bill to just under $599 million. The judge said the tab will probably increase after the appeals of both companies are resolved.

Apple is seeking more damages and Samsung a complete dismissal of the case in the U.S. Court of Appeals for the Federal Circuit, the Washington, D.C.-based court that handles all patent appeals. The new trial to recalculate the damages could also increase the award.

Still, the ruling was the second significant setback in Koh’s courtroom since the headline grabbing verdict was announced.

In December, Koh refused to order a sales ban on the products the jury found infringed Apple’s patents. She said Apple failed to prove the purloined technology is what drove consumers to buy a Samsung product instead of an Apple iPhone or iPad. Samsung says that it is continues to sell only three of the two dozen products found to have infringed Apple’s patents.

After a three-week trial closely followed in Silicon Valley, the jury decided that Samsung ripped off the trailblazing technology and sleek designs used by Apple to create its revolutionary iPhone and iPad. Jurors ordered Samsung to pay Apple $1.05 billion.

Apple filed another lawsuit last year accusing Samsung’s newer line of products of continuing to use technology controlled by Apple. Koh has scheduled trial in that case for early next year. She has implored both companies on several occasions to settle their difference with little success.

Apple filed its patent infringement lawsuit in April 2011 and engaged legions of the country’s highest-paid patent lawyers to demand $2.5 billion from its top smartphone competitor. Samsung Electronics Co. fired back with its own lawsuit seeking $399 million.

The jury found that several Samsung products illegally used such Apple creations as the “bounce-back” feature when a user scrolls to an end image, and the ability to zoom text with a tap of a finger.

Samsung has mounted an aggressive post-trial attack on the verdict, raising a number of legal issues that allege the South Korean company was treated unfairly in a federal courtroom a dozen miles from Apple’s Cupertino headquarters. Samsung alleges that some of Apple’s patents shouldn’t have been awarded in the first place and that the jury made mistakes in calculating the damage award.

Samsung has emerged as one of Apple’s biggest rivals and has overtaken it as the leading smartphone maker. Samsung’s Galaxy line of phones run on Android, a mobile operating system that Google Inc. has given out for free to Samsung and other phone makers.

Apple and Samsung have filed similar lawsuits in eight other countries, including South Korea, Germany, Japan, Italy, the Netherlands, Britain, France and Australia.