Abstract. The most popular method for computing the matrix logarithm is the inverse scaling and squaring method, which is the basis of the recent algorithm of [A. H. Al-Mohy and N. J. Higham, Improved inverse scaling and squaring algorithms for the matrix logarithm, SIAM J. Sci. Comput., 34 (2012), pp. C152-C169]. We show that by differentiating the latter algorithm a backward stable algorithm for computing the Fréchet derivative of the matrix logarithm is obtained. This algorithm requires complex arithmetic, but we also develop a version that uses only real arithmetic when A is real; as a special case we obtain a new algorithm for computing the logarithm of a real matrix in real arithmetic. We show experimentally that our two algorithms are more accurate and efficient than existing algorithms for computing the Fréchet derivative. We also show how the algorithms can be used to produce reliable estimates of the condition number of the matrix logarithm.Key words. matrix logarithm, principal logarithm, inverse scaling and squaring method, Fréchet derivative, condition number, Padé approximation, backward error analysis, matrix exponential, matrix square root, MATLAB, logm.
Abstract. Several existing algorithms for computing the matrix cosine employ polynomial or rational approximations combined with scaling and use of a double angle formula. Their derivations are based on forward error bounds. We derive new algorithms for computing the matrix cosine, the matrix sine, and both simultaneously that are backward stable in exact arithmetic and behave in a forward stable manner in floating point arithmetic. Our new algorithms employ both Padé approximants of sin x and new rational approximants to cos x and sin x obtained from Padé approximants to e x . The amount of scaling and the degree of the approximants are chosen to minimize the computational cost subject to backward stability in exact arithmetic. Numerical experiments show that the new algorithms have backward and forward errors that rival or surpass those of existing algorithms and are particularly favorable for triangular matrices.
ObjectiveEstimating survival can aid care planning, but the use of absolute survival projections can be challenging for patients and clinicians to contextualise. We aimed to define how heart failure and its major comorbidities contribute to loss of actuarially predicted life expectancy.MethodsWe conducted an observational cohort study of 1794 adults with stable chronic heart failure and reduced left ventricular ejection fraction, recruited from cardiology outpatient departments of four UK hospitals. Data from an 11-year maximum (5-year median) follow-up period (999 deaths) were used to define how heart failure and its major comorbidities impact on survival, relative to an age–sex matched control UK population, using a relative survival framework.ResultsAfter 10 years, mortality in the reference control population was 29%. In people with heart failure, this increased by an additional 37% (95% CI 34% to 40%), equating to an additional 2.2 years of lost life or a 2.4-fold (2.2–2.5) excess loss of life. This excess was greater in men than women (2.4 years (2.2–2.7) vs 1.6 years (1.2–2.0); p<0.001). In patients without major comorbidity, men still experienced excess loss of life, while women experienced less and were non-significantly different from the reference population (1 year (0.6–1.5) vs 0.4 years (−0.3 to 1); p<0.001). Accrual of comorbidity was associated with substantial increases in excess lost life, particularly for diabetes, chronic kidney and lung disease.ConclusionsComorbidity accounts for the majority of lost life expectancy in people with heart failure. Women, but not men, without comorbidity experience survival close to reference controls.
The worldwide prevalence of Parkinson's disease is increasing. There is urgent need for new tools to objectively measure the condition. Existing methods to record the cardinal motor feature of the condition, bradykinesia, using wearable sensors or smartphone apps have not reached large-scale, routine use. We evaluate new computer vision (artificial intelligence) technology, DeepLabCut, as a contactless method to quantify measures related to Parkinson's bradykinesia from smartphone videos of finger tapping. MethodsStandard smartphone video recordings of 133 hands performing finger tapping (39 idiopathic Parkinson's patients and 30 controls) were tracked on a frame-by-frame basis with DeepLabCut. Objective computer measures of tapping speed, amplitude and rhythm were correlated with clinical ratings made by 22 movement disorder neurologists using the Modified Bradykinesia Rating Scale (MBRS) and Movement Disorder Society revision of the Unified Parkinson's Disease Rating Scale (MDS-UPDRS). ResultsDeepLabCut reliably tracked and measured finger tapping in standard smartphone video. Computer measures correlated well with clinical ratings of bradykinesia (Spearman coefficients): -0.74 speed, 0.66 amplitude, -0.65 rhythm for MBRS; -0.56 speed, 0.61 amplitude, -0.50 rhythm for MDS-UPDRS; 0.69 combined for MDS-UPDRS. All p < 0.001. ConclusionNew computer vision software, DeepLabCut, can quantify three measures related to Parkinson's bradykinesia from smartphone videos of finger tapping. Objective 'contactless' measures of standard clinical examinations were not previously possible with wearable sensors (accelerometers, gyroscopes, infrared markers).DeepLabCut requires only conventional video recording of clinical examination and is entirely 'contactless'. This next generation technology holds potential for Parkinson's and other neurological disorders with altered movements.
Abstract. The Fréchet derivative L f of a matrix function f : C n×n → C n×n controls the sensitivity of the function to small perturbations in the matrix. While much is known about the properties of L f and how to compute it, little attention has been given to higher order Fréchet derivatives. We derive sufficient conditions for the kth Fréchet derivative to exist and be continuous in its arguments and we develop algorithms for computing the kth derivative and its Kronecker form. We analyze the level-2 absolute condition number of a matrix function ("the condition number of the condition number") and bound it in terms of the second Fréchet derivative. For normal matrices and the exponential we show that in the 2-norm the level-1 and level-2 absolute condition numbers are equal and that the relative condition numbers are within a small constant factor of each other. We also obtain an exact relationship between the level-1 and level-2 absolute condition numbers for the matrix inverse and arbitrary nonsingular matrices, as well as a weaker connection for Hermitian matrices for a class of functions that includes the logarithm and square root. Finally, the relation between the level-1 and level-2 condition numbers is investigated more generally through numerical experiments.Key words. matrix function, Fréchet derivative, Gâteaux derivative, higher order derivative, matrix exponential, matrix logarithm, matrix square root, matrix inverse, matrix calculus, partial derivative, Kronecker form, level-2 condition number, expm, logm, sqrtm, MATLAB AMS subject classifications. 65F30, 65F60DOI. 10.1137/130945259 Introduction. Matrix functions f : Cn×n → C n×n such as the matrix exponential, the matrix logarithm, and matrix powers A t for t ∈ R are being used within a growing number of applications including model reduction [5] The Fréchet derivative of f at A ∈ C n×n is the unique function L f (A, · ) that is linear in its second argument and for all E ∈ C n×n satisfies
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