+- HP Forums (https://www.hpmuseum.org/forum)
+-- Forum: HP Calculators (and very old HP Computers) (/forum-3.html)
+--- Forum: General Forum (/forum-4.html)
+--- Thread: 15C/LE/CE etc (/thread-20329.html)
15C/LE/CE etc - SlideRule - 08-17-2023 12:00 AM
An interesting post on RadioExperimenter by Paul J. McClelian.
Hp15c Calculator
1 0 0.3333333333 which is the LU decomposition of the singular matrix 3 3 0.9999999999 0.9999999999
The calculated determinants of A and C are also identical.
Because the calculated LU decompositions of some singular and nonsingular matrices are identical, any test for singularity based upon a calculated decomposition would be unreliable. Some singular matrices would fail the test and some nonsingular ones would pass it. Therefore, no such test is built into the HP-15C.
Instead, if a calculated diagonal element of U. which we call a pivot, is found to be zero during the LU decomposition, rather than aborting the matrix calculation and reporting the input matrix to be singular, the HP-15C replaces the zero pivot by a small positive number and continues with the calculation. This number is usually small compared to the rounding errors in the calculations. Specifically. it will be about 10-11 times the largest absolute value of any element in that column of the original matrix. If every element in that column of the original matrix has an absolute value less than 10E-89, the value 10E-99" is used instead.
An advantage of replacing zero pivots by nonzero pivots is that matrix inversion and system solution calculations will not be interrupted by zero pivots. This is especially useful in applications such as calculating eigenvectors using the method of inverse iteration. Example programs calculating eigenvalues and eigenvectors can be found in reference 3.
The effect of rounding errors and possible intentional perturbations causes the calculated decomposition to have all nonzero pivots and to correspond to a nonsingular matrix usually identical to or negligibly different from the original matrix.
Complex Matrix Calculations
The HP-15C only operates on real matrices, that is, matrices with real elements. However, it is possible to represent complex matrices as real matrices and to perform matrix addition, subtraction, multiplication, and inversion of complex matrices and to solve complex systems of equations using these real representations.
Let Z = X + iY denote a complex matrix with real part X and imaginary part Y. both real matrices. One way to represent Z as a real matrix isas the partitioned matrix -m having twice the number of rows but the same number of columns as Z, Complex matrices can be added or subtracted by adding and subtracting such real representations.
Another computationally useful real representation for Z having twice the number of rows, and columns as Z. The HP-l 5C*s built-in matrix operation MATRIX 2 performs the transformation The operation MATRIX 3 performs the inverse transformation Z zf
Suppose A. B and C are complex matrices and A is invertible. Then complex matrix multiplication, inversion, and system solution can be performed with real matrices and built-in HP-15C operations using the relationships; These procedures are illustrated in the HP-15C Owner's Handbook.
Matrix Transpose
The operations MATRIX 2 and MATRIX 3 perform their transformations using a matrix transpose routine. The rows and columns of a matrix are interchanged to form the transpose of that matrix. The transformation is performed in place, replacing the original matrix by its transpose. This routine is available to the user as MATRIX 4. Consider the following example:
Here the elements of the matrices have been displayed in a two-dimensional format. However, they are stored in a one-dimensional sequence within the calculator's memory. For this example, the transpose operation changes the ordering of the elements within the calculator memory as abcdef-*adbecf. The MATRIX 4 operation moves the elements according to
These movements form disjoint loops. The first value in the sequence is the first candidate for moving. As a value is copied into its destination, that destination is tagged in its XS field. The previous value at that location is the next candidate for moving. Movement along a loop continues until a destination is encountered that is already tagged. The content of the tagged destination is not changed and the current loop is terminated. The value in the location immediately following that tagged destination is the next candidate for moving.
This operation continues moving values along loops until the sequence is exhausted, at which point all destination tags are removed. Finally, the recorded dimensions of the matrix are switched.
Accuracy of Matrix Calculations
Accuracy specifications for all matrix operations are given in reference 3. These specifications are stated in terms of both backward and forward error analysis. Reference 3 includes a general rule of thumb for the number of significant digits in a calculated matrix inverse or system solution. It also includes descriptions of techniques to improve upon the accuracy of calculated system solutions and to reduce the ill-conditioning of systems of equations.
Acknowledgments
Numerous individuals made valuable contributions to the HP-15C software effort. As the software project manager, Rich Carone helped formulate some of the original design concepts and kept the software effort on track. Diana Roy, Robert Barkan, and Hank Schroeder wrote the HP-15C Owner's Handbook. We would like to give special thanks to Professor William Kahan, who contributed many design ideas, provided strong guidance in developing the mathematical algorithms, and wrote a portion of the HP-15C Advanced Functions Handbook. His unbounded enthusiasm for the product helped keep us going, especially when we still had features to implement and no ROM space left.
the original article is not difficult to find.
BEST!
SlideRule
OCR correction noted
RE: 15C/LE/CE etc - EdS2 - 08-17-2023 07:24 AM
Looks like this is extracted from May 1983 HP Journal specifically the article Scientific Pocket Calculator Extends Range of Built-in Functions, by Eric A. Even, Paul J. McClellan, and Joseph P. Tanzini
Correcting an OCR error:
Quote:If every element in that column of the original matrix has an absolute value less than 10E-89, the value 10E-99 is used instead.
Edit to add: the following article in the same issue is on the HP-16C: A Pocket Calculator for Computer Science Professionals, by Eric A. Evet