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Functions involving two or more variables require multidimensional array indexing techniques. The latter case may thus employ a two-dimensional array of power[x][y] to replace a function to calculate x y for a limited range of x and y values. Functions that have more than one result may be implemented with lookup tables that are arrays of ...
Use of a user-defined function sq(x) in Microsoft Excel. The named variables x & y are identified in the Name Manager. The function sq is introduced using the Visual Basic editor supplied with Excel. Subroutine in Excel calculates the square of named column variable x read from the spreadsheet, and writes it into the named column variable y.
Formulas in the B column multiply values from the A column using relative references, and the formula in B4 uses the SUM() function to find the sum of values in the B1:B3 range. A formula identifies the calculation needed to place the result in the cell it is contained within. A cell containing a formula, therefore, has two display components ...
The main idea of the method is to partition the matrix into small square blocks of size t × t for some parameter t, and to use a lookup table to perform the algorithm quickly within each block.
Rule: to multiply by 12: Starting from the rightmost digit, double each digit and add the neighbor. (The "neighbor" is the digit on the right.) If the answer is greater than a single digit, simply carry over the extra digit (which will be a 1 or 2) to the next operation. The remaining digit is one digit of the final result.
First multiply the quarters by 47, the result 94 is written into the first workspace. Next, multiply cwt 12*47 = (2 + 10)*47 but don't add up the partial results (94, 470) yet. Likewise multiply 23 by 47 yielding (141, 940). The quarters column is totaled and the result placed in the second workspace (a trivial move in this case).
The elementary functions are constructed by composing arithmetic operations, the exponential function (), the natural logarithm (), trigonometric functions (,), and their inverses. The complexity of an elementary function is equivalent to that of its inverse, since all elementary functions are analytic and hence invertible by means of Newton's ...
Left multiplying both by P, AP = PD. Each column of P must therefore be an eigenvector of A whose eigenvalue is the corresponding diagonal element of D. Since the columns of P must be linearly independent for P to be invertible, there exist n linearly independent eigenvectors of A.