text: "Industrial-strength" extremists cannot be dealt with "just with force of arms", he said in a rare interview.
  During the Iraq war, Mr Petraeus devised the strategy that saw a "surge" in US troop numbers and secured support from Sunni tribesmen against al-Qaeda.
  Iraq's US-backed army is now battling to retake territory seized by IS.
  Gen Petraeus described the group as "a formidable enemy".
  "It is really a conventional army that also has elements of an insurgency, and indeed significant terrorist elements as well," he said.
  But when asked to compare IS with its predecessor, al-Qaeda in Iraq - which Gen Petraeus was instrumental in defeating - he said the latter "had much greater roots in Iraq and much greater numbers than IS".
  The retired general characterised the recent capture by IS of Ramadi, the capital of Anbar province, as "a strategic loss in the sense that the narrative of IS being on the defensive - of losing - was shown to be somewhat hollow".
  "I do think Ramadi will be retaken in a matter of weeks or less," he added.
  "But this was a big setback. At such a time, one has to look at the strategy, at refinements that need to be made, at efforts that need to be augmented, and I know that's what's going on right now."
  1974: Graduates from West Point US military academy, joining the army
  2007: Leads US troop surge during Iraq conflict
  2008: Head of US Central Command
  2010: Nato commander in Afghanistan
  Mid-2011: Leaves military to become CIA director
  November 2012: Resigns over affair
  After commanding international troops in Iraq and Afghanistan, Gen Petraeus became head of the CIA but was forced to stand down in 2012, following revelations of an extra-marital affair.
  Subsequent allegations that he provided classified information to his mistress while he was CIA director led him to plead guilty to one misdemeanour charge.
  But despite the turbulence of his private life, Gen Petraeus remains one of the heroes of America's controversial military operations of recent years.
  He was the architect of the "surge" in Iraq that deployed more US troops, but equally important, got them out of their defended compounds to create security at a local level.
  He also greatly expanded the Anbar Awakening, the mobilisation of Sunni tribesmen to combat al-Qaeda.
  Not surprisingly in the wake of the recent setbacks in Iraq, his views are being canvassed by the White House, the Pentagon, and on Capitol Hill.
  The keynote of the Petraeus approach today is as it always was - the need for the political and military aspects of strategy to march closely in step.
  "You cannot deal with an industrial-strength extremist problem just with force of arms," he said.  "You have to have that political component as well."
  Political change has to start at the top. Above all, Gen Petraeus says, "the Sunni Arabs have to be given incentives to support the new Iraq rather than to oppose it".
  As to the fundamental question - can the Iraqi military actually win against Islamic State? - he has few doubts.
  "During the surge and in the years after the surge, Iraqi forces fought and died for their country at vastly higher numbers than did US and coalition forces. We know that they can fight," he insists.
  "We know that they will fight. But they will only fight if they have good leadership, and the support and knowledge that somebody will have their back if they get into a tough fight."
  That looks like requiring greater US involvement closer to the frontline. Gen Petraeus is reluctant to give details.
  "Should there be US advisors with Iraqi forces below the divisional  level?" he asks. "Should there be joint teams of tactical air controllers on the ground with security and other assets to support them?
  "Does there need to be an augmentation of the train-and-equip effort? Can we accelerate the delivery of some of the equipment that Iraq so desperately needs?"
  It sounds like the elements of a blueprint for a re-invigorated US effort. But then again - typical Petraeus - there is the political dimension.
  "Are we doing all that we can to empower and support those Iraqi leaders, starting with Prime Minister Haider al-Abadi, who recognise the need to bring the Sunni Arab population back into the fabric of Iraqi society? And, most importantly, are our military elements and structures sufficiently supporting the political component and vice versa?"
  In response to my comment that all this could have been said a year ago - Gen Petraeus responded by saying that he did indeed say it all a year ago. This is not about re-inventing the wheel.
  "We need to see the same elements of this (previous) comprehensive civil-military counter- insurgency campaign, albeit today with the Iraqis playing the key roles, enabled by the Americans."
  Gen Petraeus acknowledges the difficulties of grappling with the horrors in Syria but insists that the only way forward is to train and equip moderate opposition elements. He says that more such fighters will spring up once the programme gets under way.
  And he is very cautious about any Iranian role in combating IS. He insists Iran remains a revolutionary power in the region.
  "This is a country that has made progress because of chaos. It has both benefited from chaos and fomented chaos to try to achieve regional hegemony," he says.
  So he believes that "while there can indeed be some coincidence of interest between the US, its coalition allies in the region and Iran in terms of the defeat of IS (...) the over-arching context is one that gives you reason to have considerable caution in how you go forward in the relationship with Iran".
  Gen Petraeus still travels to Iraq regularly. He knows the key players well. And though out of uniform and established in a civilian career as an adviser, he remains the man who did achieve a kind of success in Iraq.
  That is a commodity that is in short supply today for the Americans and the Iraqi authorities. This  makes the Petraeus approach an attractive model.
  But the question remains: Can the outcome be the same with the Iraqi military cast in the role of the Americans?
target: Ex-CIA Director David Petraeus has told the BBC that Islamic State militants can only be defeated through a dual military and political approach.