The most rigorous study to date of how much it costs to care for Americans with dementia found that the financial burden is at least as high as that of heart disease or cancer, and is probably higher. And both the costs and the number of people with dementia will more than double within 30 years, skyrocketing at a rate that rarely occurs with a chronic disease. The research, led by an economist at the RAND Corporation, financed by the federal government, and published Wednesday in The New England Journal of Medicine, provides the most reliable basis yet for measuring the scale of the problem.