Efficient Construction, Update and Downdate Of The Coefficients Of Interpolants Based On Polynomials Satisfying A Three-Term Recurrence Relation

In this paper, we consider methods to compute the coefficients of interpolants relative to a basis of polynomials satisfying a three-term recurrence relation. Two new algorithms are presented: the first constructs the coefficients of the interpolation incrementally and can be used to update the coefficients whenever a nodes is added to or removed from the interpolation. The second algorithm, which constructs the interpolation coefficients by decomposing the Vandermonde-like matrix iteratively, can not be used to update or downdate an interpolation, yet is more numerically stable than the first algorithm and is more efficient when the coefficients of multiple interpolations are to be computed over the same set of nodes.Comment: 18 pages, submitted to the Journal of Scientific Computing

Robust Padé approximation via SVD

Padé approximation is considered from the point of view of robust methods of numerical linear algebra, in particular the singular value decomposition. This leads to an algorithm for practical computation that bypasses most problems of solution of nearly-singular systems and spurious pole-zero pairs caused by rounding errors; a Matlab code is provided. The success of this algorithm suggests that there might be variants of Padé approximation that would be pointwise convergent as the degrees of the numerator and denominator increase to infinity, unlike traditional Padé approximants, which converge only in measure or capacity

Hash comparisons to provide warnings of phishing attacks

This disclosure describes techniques to warn users when a password for a particular online domain is being submitted by a user device to a domain different than the one for which it was originally registered. The warning alerts the user of possible phishing attacks. Cryptographic hashes of user passwords along with corresponding domains are stored. When a user attempts to send data to a website, e.g., via form submission, hashes of the data are calculated and compared with the stored hashes of password. In the case of a match, a warning is presented to the user. The techniques can be implemented as a feature of a web browser, a browser plugin, as standalone software, as part of an operating system, etc

Automatic obfuscation of sensitive content on locked devices

The content of notifications from the device operating system and applications is displayed on the screen of a device even when the device is locked. Such notifications make private or sensitive information accessible by other parties without unlocking the device. To counter this risk, device operating systems allow users to hide notification details when the device is locked. However, such hiding results in significant loss of understandability and utility of notifications. Per techniques of this disclosure, a trained machine learning based model is applied with specific user permission to identify sensitive content by analyzing pixels of the device lock screen image that includes notifications. The techniques may be incorporated within the device operating system or web application frameworks. If the inferred level of sensitivity is higher than a threshold value, the corresponding content is obfuscated when displaying the notification on a locked device