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Quadruple Integral Calculator Real 15 for Windows 2000, Windows XP, Vista, and Windows 7
This calculator calculates quadruple definite integrals utilizing Tanh-sinh quadrature algorithm. The algorithm does not utilize any symbolic integration. It requires roughly the same time for integration of e^cos(x+y+z+w) and cox(x+y+z+w), although the first is not symbolically integrable. Integration of cos(x+y+z+w) on [-2, 2]X[-1,1]X[-1,1]X[-1,1] is twice longer than integration of cos(x+y+z+w) on [-1, 1]X[-1,1]X[-1,1]X[-1,1].
In general the precision of calculations is about 1E-15 (1E-16 for simple functions). In practice the precision of calculation greatly varies and depends on functions. We recommend to start with low precision and watch conversion. If conversion is quick then it is safely to increase precision.
The region (or domain) of integration in xyzw-space is given by numerical limits for x, lower and upper contours for y given as functions of x, lower and upper surfaces for z given as functions of x,y, and lower and upper hypersurfaces for w given as functions of x,y,z. For example, lower limit of x = -1, upper limit of x = 1, lower contour of y = -(1-x^2)^0.5, upper contour of y = (1-x^2)^0.5, lower surface -(1-x^2-y^2)^0.5, upper surface (1-x^2-y^2)^0.5, lower hypersurface -(1-x^2-y^2-z^2)^0.5, upper surface (1-x^2-y^2-z^2)^0.5 define four-dimensional ball of radius one centered at origin. The contours and surfaces need not to be closed (endpoints coincide). For example, [-2,2] for x, lower contour = -cos(x), upper contour = cos(x). If contours intersect in the middle, then the resulting integral is a sum of positive (counterclockwise path) and negative (clockwise path) integrals. Similarly for surfaces.
This calculator computes definite Quadruple integrals by Quadruple tinh-sinh quadrature scheme.
Type formula into Integrand text-box. Change default values of lower and upper limits of integration. Click Calculate button. Watch if the algorithm is converging. The current uncertainty value should be steadily decreasing. If the algorithm is diverging, click any of Stop buttons. Observe results of calculation in History tab.
At each level the number of calculation points doubles and so does the number of accurate digits. You can watch it by decreasing exponent of current uncertainty. The number of points of calculation depends on level and length of interval of integration for tinh-sinh quadrature algorithm.
If you use in integrand functions with accuracy lower than the default for the calculator, then adjust Desired Uncertainty value.
The convergence of algorithm is not guaranteed for all functions and all intervals of integration. But even worse, it is possible to construct a function for which the algorithm gets zero at level 1 and Real 15 (zeroes at points of calculation) and thus successfully terminates, although the actual integral is not zero. For example, (sin(8**asinh(2/*atanh(x))))^2 . Fortunately, such examples are artificial and rare.
Conclusion. Tanf-sinh quadrature scheme is fast and and gives right answer for large variety of functions. But for some functions and intervals the algorithm diverges. And for some functions the scheme gives wrong answer.
The easiest way to edit formula is left-clicking buttons. It allows to keep brackets balanced, functions names correct and so on. Clicking the button "calculate" triggers calculation of entered formula. The result of calculation appears in the window (text-box) named Result.
The second way is to use keyboard (and keypad). All controls usual for editing are available. Pressing the key Enter triggers calculation. Before using keyboard don't forget to click inside text-box to get focus (blinking cursor).
After calculation the entered formula is not deleted from the Edit window allowing to modify formula. If you want to delete formula select it by mouse and delete. For selecting text you can use right-click menu "Select All" or left-click mouse dragging along the text. For deleting selected text use right-click menu "Cut" or "Delete".
Using right-click menu you can copy and paste text between Edit window and all other text-box windows.
For copying text from saved History file open saved History file (usually in WordPad, Notepad, or MS Word ), drag mouse along the text for selection and then choose Copy from right-click menu. Then go to Formula tab, right-click onto Edit window, select command Paste.
Apply the same procedure for copying text from History window or saved History file into variables windows in Variables tab.
Functions and operations have to be entered exactly as they appears by pressing buttons. Alternative names are not supported.
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