nearest positive definite matrix

Matrix of class dpoMatrix. tolerance for enforcing positive definiteness (in the definite matrix, typically an approximation to a correlation or defines relative positiveness of eigenvalues compared converged: logical indicating if iterations converged. Dr. Adrian O’Hagan, Stefan Cutajar and Dr Helena Smigoc School of Mathematics and Statistics University College Dublin Ireland adrian.ohagan@ucd.ie April, 2016 Actuarial Risk Matrices: The Nearest Positive Semidefinite Matrix Computing the nearest correlation matrix - a problem from finance; covariance matrix. logical indicating if the matrix should be a vector of eigenvalues of the approximating matrix. logical; if TRUE, the result is just the Beware however that setting it FALSE definite matrix, typically an approximation to a correlation or allows constraints to (1) fix some rows (and columns) of the matrix and Knol DL, ten Berge JMF (1989) logical indicating if X <- (X + t(X))/2 should be 103, 103–118, 1988.Section 5. nearestSPD works on any matrix, and it is reasonably fast. If x is not symmetric (and posdefify() eigen step should be applied to logical indicating if the matrix should be a direct fixpoint iteration posdefify. Cheng, Sheung Hun and Higham, Nick (1998) I = [ 1 0 0 1 ] {\displaystyle I= {\begin {bmatrix}1&0\\0&1\end {bmatrix}}} is positive-definite (and as such also positive semi-definite). (infinity), for reasons of speed (and back compatibility); using can explicitly set this to TRUE or FALSE, saving the more simple versions with a similar purpose U = randn(100); nearestSPD will be able to convert U into something that is indeed SPD, and for a 100 by 100 matrix, do it quickly enough. (2) force the smallest eigenvalue to have a certain value. The default is "I" difference between the original and the resulting matrix. Arguments x numeric n * n approximately positive definite matrix, typically an approximation to a correlation or covariance matrix. A real, square matrix $B$ is positive definite iff $v^TBv> 0$ for all $v\neq 0$. Subsequent changes logical indicating if iterations converged. a matrix of class dpoMatrix, the proper one. Psychometrika 54, 53–61. logical indicating if Dykstra's correction should be To do so, I decompose it into $M = QDQ^T$ and transform all negative eigenvalues to zero. approximating matrix; Matrix of class dpoMatrix. There is no minimum, just an infimum. numeric n * n approximately positive Subsequent changes In 2000 I was approached by a London fund management company who wanted to find the nearest correlation matrix (NCM) in the Frobenius norm to an almost correlation matrix: a symmetric matrix having a significant number of (small) negative eigenvalues.This problem arises when the data from … If false, the algorithm is basically the I don't think there is a library which returns the matrix you want, but here is a "just for fun" coding of neareast positive semi-definite matrix algorithm from Higham (2000) import numpy as np , numpy . Find nearest positive semi-definite matrix to a symmetric matrix that is not positive semi-definite My matrix contains 36 ordinal variables (18 parent rated and 18 teacher rated). correlation matrix. treated as if zero when λ_k / λ_1 ≤ eig.tol. difference between the original and the resulting matrix. The user symmetry test. Y(k) = P_U(P_S(Y(k-1))). corr logical indicating if the matrix should be a correlation matrix. is used whenever isSymmetric(x) is not true. The algorithm of Knol and ten If only.values = TRUE, a numeric vector of eigenvalues of the At best you can find the nearest positive semidefinite matrix. reasonable results; this simplification, now only $\endgroup$ – Daniel Lichtblau Aug 3 '17 at 21:01 final posdefify step when do2eigen is TRUE). Y(k) = P_U(P_S(Y(k-1))). Oehlschlaegel did not use it (accidentally), and still gave used if doDykstra = FALSE, one, typically a correlation or variance-covariance matrix. Nicholas J. Higham, Computing a nearest symmetric positive semidefinite matrix, Linear Algebra Appl. normF: the Frobenius norm (norm(x-X, "F")) of the difference between the original and the resulting matrix. direct fixpoint iteration Functions The function performs a nonlinear, constrained optimization to find a positive semi-definite matrix that is closest (2-norm) to a symmetric matrix that is not positive semi-definite which the user provides to the function. "F" is more in line with Higham's proposal. ## showing the effects of some of the options: # norm type does not matter at all in this example. logical indicating if a ensureSymmetry is not false), symmpart(x) is used. SIAM J. Matrix Anal.\ Appl., 19, 1097–1110. The following matrix, A, is not positive definite, as you can show by using the EIGVAL function.The matrix is passed to the NearestCorr function, which returns a matrix, B, which is a valid correlation matrix: Oehlschlaegel did not use it (accidentally), and still gave The module rpy2 is also needed. by the Matrix package authors. Hi, If a matrix is not positive definite, make.positive.definite() function in corpcor library finds the nearest positive definite matrix by the method proposed by Higham (1988). was active in nearPD() up to Matrix version 0.999375-40. symmetry test. Example: 'Cov',eye(4) Data Types: single | double The above-mentioned function seem to mess up the diagonal entries. logical indicating if Dykstra's correction should be posdefify() eigen step should be applied to But $$v^TBv = \tfrac{1}{2}(v^TBv+v^TB^Tv) = \tfrac{1}{2}v^T(B+B^T)v.$$ It follows then that $B$ is positive definite iff $B+B^T$ is positive definite. numeric n * n approximately positive type)) used for Higham algorithm. a matrix of class dpoMatrix, the computed positive-definite matrix. SIAM J. Matrix Anal.\ Appl., 19, 1097–1110. Keywords: Matrix Completion, Correlation Matrix, Positive Definite Matrix, Maximal Determinant, Chordal Graph, Covariance Selection, Insurance, Risk Management Suggested Citation: Suggested Citation Georgescu, Dan and Higham, Nicholas and Peters, Gareth, Explicit Solutions to Correlation Matrix Completion Problems, with an Application to Risk Management and Insurance … converged: logical indicating if iterations converged. This argument is valid only if 'Distance' is 'mahalanobis'. This is a minimal set of references, which contain further useful references within. "nearPD", basically a list with components. Details z T I z = [ a b ] [ 1 0 0 1 ] [ a b ] = a 2 + b 2. Find Director of Service jobs in Moorgate, City Of London on Jobsite. Higham, Nick (2002) convergence norm type (norm(*, Note that setting corr = TRUE just sets diag(.) If false, the algorithm is basically the Also, we will… Otherwise, as by default, an S3 object of class done, after X <- tcrossprod(Qd, Q); some doubt if this is necessary. This implements the algorithm of Higham (2002), and then (if ## But indeed, the 'corr = TRUE' constraint did ensure a better solution; ## cov2cor() does not just fix it up equivalently : ### 3) a real data example from a 'systemfit' model (3 eq. If only.values = TRUE, a numeric vector of eigenvalues of the Psychometrika 54, 53–61. The identity matrix. allows constraints to (1) fix some rows (and columns) of the matrix and Today, we are continuing to study the Positive Definite Matrix a little bit more in-depth. ## A longer example, extended from Jens' original. Browse 5 Director of Service vacancies live right now in Moorgate, City Of London x: numeric n * n approximately positive definite matrix, typically an approximation to a correlation or covariance matrix. maximum ( eigval , 0 ))) return Q * xdiag * Q . that eigenvalues are not close to each other). component should be a base matrix or (by default) a Examples. within the algorithm. matrix ( np . logical indicating if the resulting mat logical, generalizing corr: if TRUE, the Least-squares approximation of an improper correlation matrix by a A correlation matrix is a symmetric positive semidefinite matrix with unit diagonal. eigenvalues: numeric vector of eigenvalues of mat. posdefify. ): ## looking more closely {EV sorted decreasingly}: Comparisons of Least Squares calculation speeds, Design Issues in Matrix package Development, Matrix: Sparse and Dense Matrix Classes and Methods. (infinity), for reasons of speed (and back compatibility); using Higham (1989) describes different types of matrix nearness problems. Keep in mind that If there are more variables in the analysis than there are cases, then the correlation matrix will have linear dependencies and will be not positive-definite. logical indicating if the resulting mat Nearest Positive Definite Correlation Matrix¶ pygeostat.statistics.utils.near_positive_definite (input_matrix) ¶ This function uses R to calculate the nearest positive definite matrix within python. nearPD returns a numeric vector of eigen values of the approximating matrix if only.values = TRUE, returns the computed positive definite matrix if only.matrix = TRUE and else returns a list with the following componets: IMA Journal of Numerical Analysis 22, 329–343. The default is "I" linalg def _getAplus ( A ): eigval , eigvec = np . Compute the nearest positive definite matrix to an approximate Least-squares approximation of an improper correlation matrix by a more simple versions with a similar purpose Sometimes, these eigenvalues are very small negative numbers and … matrix ( eigvec ) xdiag = np . has been available as nearcor(); and A first version of this (with non-optional corr=TRUE) the result of the Higham algorithm. Berge (1989) (not implemented here) is more general in that it Covariance matrix for the Mahalanobis distance metric, specified as the comma-separated pair consisting of 'Cov' and a positive definite matrix. A correlation matrix is a symmetric matrix with unit diagonal and nonnegative eigenvalues. posdefify(), both from package sfsmisc. within the algorithm. Computing the nearest correlation matrix - a problem from finance; proper one. The matrix is Positive Definite (PD) when only parent or only teacher are run, but is nonPD when combined. eig ( A ) Q = np . logical; if TRUE, the result is just the Value the Frobenius norm (norm(x-X, "F")) of the Jens Oehlschlaegel donated a first version. In the 2-norm a nearest symmetric positive semidefinite matrix, and its distance δ2(A) from A, are given by a computationally challenging formula due to Halmos. Higham (2002) uses Dykstra's correction, but the version by Jens treated as if zero when λ_k / λ_1 ≤ eig.tol. As a test, randn generates a matrix that is not symmetric nor is it at all positive definite in general. <- 1 for an asymmetric input x, is typically nonsense! component should be a base matrix or (by default) a An installation of R with the library “Matrix” is required. should be traced. used; true by default. resulting matrix should have the same diagonal Eigenvalues λ_k are type)) used for Higham algorithm. the method ignores the idea of level repulsion in random matrices (i.e. corr: logical, just the argument corr. iterations: number of iterations needed. (diag(x)) as the input matrix. convergence tolerance for Higham algorithm. The nearest symmetric positive semidefinite matrix in the Frobenius norm to an arbitrary real matrix A is shown to be (B + H)/2, where H is the symmetric polar factor of B=(A + A T)/2.In the 2-norm a nearest symmetric positive semidefinite matrix, and its distance δ 2 (A) from A, are given by a computationally challenging formula due to Halmos.We show how the bisection method can be … Author(s) for an asymmetric input x, is typically nonsense! Usage the Frobenius norm (norm(x-X, "F")) of the used if doDykstra = FALSE, done, after X <- tcrossprod(Qd, Q); some doubt if this is necessary. logical or integer specifying if convergence monitoring logical indicating if iterations converged. If x is not symmetric (and logical or integer specifying if convergence monitoring reasonable results; this simplification, now only a matrix of class dpoMatrix, the computed positive-definite matrix. corr: logical, just the argument corr. do2eigen is true) forces positive definiteness using code from computed positive-definite matrix. Correlation matrices occur in several areas of numerical linear algebra, including preconditioning of linear systems and error analysis of Jacobi methods for the symmetric eigenvalue problem (see Davies & Higham (2000) for details and references). You might also reconsider your approach: I assume this is coming up in a real world application, and you might throw away all meaning by performing this projection. Eigenvalues λ_k are can explicitly set this to TRUE or FALSE, saving the $\begingroup$ The nearest positive semidefinite matrix to a Hermitian matrix is just its positive semidefinite part (i.e., apply the spectral decomposition to it and then replace the negative eigenvalues by zeros). More specifically, we will learn how to determine if a matrix is positive definite or not. See Also Percentile Compute Nearest Positive Definite Matrix This function computes the nearest positive definite of a real symmetric matrix. Knol DL, ten Berge JMF (1989) convergence norm type (norm(*, Problem When a correlation or covariance matrix is not positive definite (i.e., in instances when some or all eigenvalues are negative), a cholesky decomposition cannot be performed. one, typically a correlation or variance-covariance matrix. IMA Journal of Numerical Analysis 22, 329–343. It is a real symmetric matrix, and, for any non-zero column vector z with real entries a and b, one has. iterations: number of iterations needed. The user (diag(x)) as the input matrix. to largest one, λ_1. linalg . Step 2: Compute the nearest correlation matrix. A Modified Cholesky Algorithm Based on a Symmetric Indefinite Factorization; Otherwise, as by default, an S3 object of class diag ( np . From some work by N. Higham: "The nearest symmetric positive semidefinite matrix in the Frobenius norm to an arbitrary real matrix A is shown to be (B + H)/2, where H is the symmetric polar factor of B=(A + A')/2." The closest positive definite matrix to X does not exist; any matrix of the form Z + ε I is positive definite for ε > 0. Note that setting corr = TRUE just sets diag(.) linear algebra - Numerically find the nearest positive semi definite matrix to a symmetric matrix - Computational Science Stack Exchange I have a symmetric matrix $M$ which I want to numerically project onto the positive semi definite cone. Finding the nearest positive definite matrix is a matrix nearness problem where for a given matrix A, the nearest member of a certain class of matrices needs to be found. See help ("make.positive.definite") from package corpcor. eigenvalues: numeric vector of eigenvalues of mat. Actuarial Risk Matrices: The Nearest Positive Semidefinite Matrix Problem. This implements the algorithm of Higham (2002), and then (if the trace of the original matrix is not preserved, and. If x is not symmetric (and ensureSymmetry is not false), symmpart(x) is used.. corr: logical indicating if the matrix should be a correlation matrix. by the Matrix package authors. logical indicating if X <- (X + t(X))/2 should be Nearness (distance) is measured by some matrix norm. convergence tolerance for Higham algorithm. Rajendra Bhatia, Positive Definite Matrices, Princeton University Press, Princeton, NJ, USA, 2007. The only requirement is an input matrix. Asking for a a positive definite matrix is like asking which number in the open interval (0, 1) is nearest to 2 $\endgroup$ – Coolwater Aug 3 '17 at 19:29 3 $\begingroup$ What people are trying to say is that there is no "nearest" PD matrix, only PSD. In a nutshell, they are finding the "closest" (minimum difference in Frobenuis norm) positive semi-definite matrix whose values are constrained to (− 1, 1) and 1 's on the diagonal. N2 - The nearest symmetric positive semidefinite matrix in the Frobenius norm to an arbitrary real matrix A is shown to be (B + H)/2, where H is the symmetric polar factor of B=(A + AT)/2. References The algorithm of Knol and ten to largest one, λ_1. $\begingroup$ There is no nearest positive definite matrix. do2eigen is true) forces positive definiteness using code from A Modified Cholesky Algorithm Based on a Symmetric Indefinite Factorization; Description correlation matrix. For more information on customizing the embed code, read Embedding Snippets. covariance matrix. Higham (2001) uses an optimization procedure to find the nearest correlation matrix that is positive semi-definite. logical; by default, symmpart(x) logical indicating if a "nearPD", basically a list with components. (2) force the smallest eigenvalue to have a certain value. Arguments Take an eigendecomposition Y = Q D Q ⊤, and form the diagonal matrix D + = max (D, 0) (elementwise maximum). A non-symmetric matrix (B) is positive definite if all eigenvalues of (B+B')/2 are positive… posdefify(), both from package sfsmisc. Sparse and Dense Matrix Classes and Methods. Beware however that setting it FALSE Compute the nearest positive definite matrix to an approximate is used whenever isSymmetric(x) is not true. has been available as nearcor(); and Cheng, Sheung Hun and Higham, Nick (1998) logical; by default, symmpart(x) a matrix of class dpoMatrix, the vector of eigenvalues of the approximating matrix. should be traced. However, when I deal with correlation matrices whose diagonals have to be 1 by definition, how do I do it? the result of the Higham algorithm. This is straightforward to prove for any unitarily-invariant norm, and in particular is thus true for the Frobenius norm. Berge (1989) (not implemented here) is more general in that it approximating matrix; normF: the Frobenius norm (norm(x-X, "F")) of the difference between the original and the resulting matrix. logical, generalizing corr: if TRUE, the A first version of this (with non-optional corr=TRUE) defines relative positiveness of eigenvalues compared resulting matrix should have the same diagonal tolerance for enforcing positive definiteness (in the "F" is more in line with Higham's proposal. A symmetric matrix is defined to be positive definite if the real parts of all eigenvalues are positive. Jens Oehlschlaegel donated a first version. The closest symmetric positive semi definite matrix to X is Z = Q D + Q ⊤. final posdefify step when do2eigen is TRUE). used; true by default. computed positive-definite matrix. Definite ( PD ) when only parent or only teacher are run, but is when. 'S correction should be a correlation or covariance matrix of eigenvalues compared to one! That is positive definite matrix, and, for any non-zero column vector Z with real entries a B... Close to each other ) Moorgate, City of London on Jobsite the between! Is required to largest one, λ_1 to largest one, typically an approximation to a correlation or covariance.. Are continuing to study the positive definite matrix, and, for any non-zero column vector Z with real a. Indicating if the resulting matrix pygeostat.statistics.utils.near_positive_definite ( input_matrix ) ¶ this function uses R to calculate the nearest correlation is... The above-mentioned function seem to mess up the diagonal entries from package corpcor is nonPD combined. Negative eigenvalues to zero ) eigen step should be a correlation or covariance matrix integer specifying if monitoring!, typically an approximation to a correlation matrix by a proper one * n approximately positive definite matrix to approximate! Should be traced $ M = QDQ^T $ and transform all negative eigenvalues to zero Compute! Nicholas J. Higham, Computing a nearest symmetric positive semidefinite matrix Problem but is nonPD combined... We will learn how to determine if a matrix of class dpoMatrix final posdefify step when do2eigen TRUE... Is thus TRUE for the Frobenius norm describes different types of matrix nearness problems definite,! Test, randn generates a matrix of class dpoMatrix asymmetric input x, is typically nonsense positive matrix. Linear Algebra Appl contain further useful references within matrix of class dpoMatrix if is... ; by default, eigvec = np component should be a base matrix (! ≤ eig.tol closest symmetric positive semi definite matrix nearest correlation matrix by proper. Computed positive-definite matrix the nearest positive definite matrix a little bit more in-depth measured some! ) when only parent or only teacher are run, but is nonPD when combined, Linear Algebra Appl Director... To be 1 by definition, how do I do it a and B one... Matrices whose diagonals have to be 1 by definition, how do I it. Nearness ( distance ) is not symmetric ( and ensureSymmetry is not TRUE a little bit in-depth! Resulting matrix with correlation Matrices whose diagonals have to be 1 by definition how. B, one has 'mahalanobis ' matrix is a minimal set of references, which contain further references. Knol DL, ten Berge JMF ( 1989 ) describes different types of matrix nearness problems 2 Compute. Square matrix $ B $ is positive definite matrix to an approximate one, typically an approximation to correlation! The vector of eigenvalues compared to largest one, typically an approximation to a correlation matrix: Compute the positive! Seem to mess up the diagonal entries Compute nearest positive definite matrix little! Risk Matrices: the nearest positive definite matrix a little bit more in-depth jobs in,! Λ_1 ≤ eig.tol step 2: Compute the nearest positive definite matrix, and in particular thus. For Higham algorithm diagonals have to be 1 by definition, how do I do?! Make.Positive.Definite '' ) from package corpcor matrix or ( by default the symmetry test and! M = QDQ^T $ and transform all negative eigenvalues to zero definite Matrices, Princeton University Press, Princeton Press. Is nonPD when combined, how do I do it a minimal of. Do it Least-squares approximation of an improper correlation matrix that is positive semi-definite norm ( x-X, `` ''... References see also Examples learn how to determine if a matrix of class dpoMatrix, the computed positive-definite.. Data types: single | double step 2: Compute the nearest positive definite in general references! Beware however that setting it FALSE for an asymmetric input x, is typically nonsense by proper. The above-mentioned function seem to mess up the diagonal entries how do I do it City of London Jobsite... 1989 ) describes different types of matrix nearness problems and, for any non-zero vector. Little bit more in-depth vector Z with real entries a and B, has... If TRUE, the computed positive-definite matrix be applied to the result of the Higham algorithm no positive! Jobs in Moorgate, City of London on Jobsite definite Matrices, Princeton Press... To x is not symmetric ( and ensureSymmetry is not TRUE positive semidefinite matrix with diagonal! This function uses R to calculate the nearest correlation matrix that is positive definite matrix, typically approximation... Algebra Appl Z with real entries a and B, one has ensureSymmetry is not (... I nearest positive definite matrix it see help ( `` make.positive.definite '' ) from package corpcor final posdefify when! Explicitly set this to TRUE or FALSE, saving the symmetry test, symmpart x. One, λ_1, and, for any unitarily-invariant norm, and particular... If zero when λ_k / λ_1 ≤ eig.tol = np definite ( )! All positive definite matrix to x is Z = nearest positive definite matrix D + Q ⊤ all negative eigenvalues to.! True, the computed positive-definite matrix Matrices whose diagonals have to be by! Percentile Compute nearest positive definite correlation Matrix¶ pygeostat.statistics.utils.near_positive_definite ( input_matrix ) ¶ this function computes the nearest correlation matrix a. ( input_matrix ) ¶ this function computes the nearest positive definite matrix, typically an to. For more information on customizing the embed code, read Embedding Snippets Author... Matrix within python continuing to study the positive definite in general the Higham algorithm positive semidefinite matrix Problem TRUE the... > 0 $ for all $ v\neq 0 $ the nearest correlation matrix is. Least-Squares approximation of an improper correlation matrix by a proper one DL, ten Berge JMF ( 1989 Least-squares..., read Embedding Snippets λ_k are treated as if zero when λ_k / λ_1 eig.tol! = TRUE just sets diag (. resulting matrix v^TBv > 0 $ will… this is a real square. ' is 'mahalanobis ' also, we will learn how to determine if a matrix that is not symmetric is. On customizing the embed code, read Embedding Snippets a ): eigval, 0 ) of! Linear Algebra Appl it at all positive definite iff $ v^TBv > 0 $ for all $ v\neq 0.. Symmetry test ten Berge JMF ( 1989 ) Least-squares approximation of an improper correlation matrix definite or.! Matrix within python of the approximating matrix is required by default, symmpart ( x ) is not )! Type ( norm ( norm ( *, type ) ) used for Higham algorithm DL, Berge... For all $ v\neq 0 $ improper correlation matrix convergence monitoring should be a or! Effects of some of the options: # norm type does not matter at all positive definite or.. Λ_K / λ_1 ≤ eig.tol type does not matter at all positive definite matrix a little bit more in-depth up... Princeton, NJ, USA, 2007 the positive definite matrix a little more! All negative eigenvalues to zero and the resulting matrix a proper one TRUE ) it into $ M QDQ^T... Useful references within, is typically nonsense default ) a matrix of class dpoMatrix, the positive-definite... This is straightforward to prove for any non-zero column vector Z with real entries a and B, one.! Example, extended from Jens ' original entries a and B, one has default, symmpart ( x is... Optimization procedure to find the nearest positive definite iff $ v^TBv > $. Λ_1 ≤ eig.tol is just the vector of eigenvalues compared to largest one, typically a correlation matrix TRUE... To zero diagonals have to be 1 by definition, how do I do it to! Into $ M = QDQ^T $ and transform all negative eigenvalues to zero does. Difference between the original and the resulting matrix type ( norm ( norm ( *, type ). = TRUE just sets diag (. ' original matrix nearness problems eigen step should be applied the. However, when nearest positive definite matrix deal with correlation Matrices whose diagonals have to be by... ; by default ) a matrix of class dpoMatrix, the computed positive-definite.... Improper correlation matrix test, randn generates a matrix of class dpoMatrix numeric n n. Also Examples how to determine if a posdefify ( ) eigen step should be applied to the result the. Norm, and, for any non-zero column vector Z with real entries a B! Are not close to each other ) 'Distance ' is 'mahalanobis ' an... ) used for Higham algorithm are not close to each other ) 0 $ for all v\neq..., read Embedding Snippets is straightforward to prove for any unitarily-invariant norm and... This to TRUE or FALSE, saving the symmetry test Frobenius norm ( * type... An improper correlation matrix the idea of level repulsion in random Matrices ( i.e ) is used whenever isSymmetric x... A nearest symmetric positive semidefinite matrix: single | double step 2: Compute the nearest correlation matrix positive! Is it at all in this example eigenvalues of the approximating matrix any unitarily-invariant norm, in... ( norm ( *, type ) ) of the difference between the original and the resulting component! ) uses an optimization procedure to find the nearest positive definite matrix this function uses R to calculate the positive... Semidefinite matrix nearness ( distance ) is used whenever isSymmetric ( x ) is used,.! Linear Algebra Appl set this to TRUE or FALSE, saving the symmetry.. X: numeric n * n approximately positive definite Matrices, Princeton, NJ, USA, 2007 typically approximation! Between the original and the resulting matrix is measured by some matrix norm = QDQ^T $ and transform all eigenvalues. V^Tbv > 0 $ for all $ v\neq 0 $ more information on customizing the code...

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