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LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares
 ACM Trans. Math. Software
, 1982
"... An iterative method is given for solving Ax ~ffi b and minU Ax b 112, where the matrix A is large and sparse. The method is based on the bidiagonalization procedure of Golub and Kahan. It is analytically equivalent to the standard method of conjugate gradients, but possesses more favorable numerica ..."
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Cited by 653 (21 self)
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An iterative method is given for solving Ax ~ffi b and minU Ax b 112, where the matrix A is large and sparse. The method is based on the bidiagonalization procedure of Golub and Kahan. It is analytically equivalent to the standard method of conjugate gradients, but possesses more favorable
DATA fffi flflfiff ffi
"... kml<npo:qsrutvruwZovnpx<lzy8{uqru}zwux<l8wZ{uovrul<{uovnv~<rqs<ruruwZzqsl8ov8rZqsnvq<{uq 8x<~<rZl ovx#y8r{qsŁ8wZwurZqsqsŁ<t%ruov<x8}#x<8x8}<Ł8wZnpl<<npT<Ł8{Ztpnvovqs8rZruwZ x8Ł<ov8Ł8ovmx<ru~Trunvlx<}8ruovx<x8}8Ł<wurm<nv<<Ł8{Ztp ..."
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{Ztpnvov%qs8rZruwu<Rtv{uTr qs<ruruwZ}<{uov{uy<{uqsrZq{urru<Ł8nvru}<.ax<qsx8%r{Z8<tpnvwZ{uovnvx8l8q.ov8nvqnvql<x8o 8{Zwuovnvwu{Zt}8Ł<rovxAov8rzwZx8%8tvru Tnvop¡x8¢op<rz}8{uov{Zy8{uqrqsru{Zwu8x<wurZqsq {ul<}4ov<r£qsovx<{uTr£rZ8Ł<nprZrZl8ovqx<qsŁ<wu¤}8
«ª®W£K§¨L¶<)¿ffiÀI§¤£n<®¤£nÁ´"§¨K°K©n´"°;¢ ¤ "<ÃÂ!Ä¿;°n´"¢·Á " "¢W´ " Å)«ª¡" §W°K°n´"¢¤°n´«ª§¤¡"<Æv·¢¤¬'°«±¨ª´ " Ç)ÈQ§I«ª£D®¤¢W§¨tÅ)§ ¤ G¡"¢x "¡«ª¬'§¤¡")´"©n£º ¡«ª¬'D^É[·ffi«ª£K·¯Q¢W£Q·Á¢nÅx
"... h"i;j^kIlnm^onj^kIp q3rtsffij^kIuvsxwyKmq;z{lKi6oK})m^y;~IkIsffis^z{ifil DoK~fikIi<ifiz{Kkm^s^z{j"rQy w1)kI~fii6yK{yKlnr )kIufi{kImffilKmffioKGkIiŁnŁ;LŁKnKŁnW"oK~fikiL'kImffiuoKi;r D'[' ..."
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h"i;j^kIlnm^onj^kIp q3rtsffij^kIuvsxwyKmq;z{lKi6oK})m^y;~IkIsffis^z{ifil DoK~fikIi<ifiz{Kkm^s^z{j"rQy w1)kI~fii6yK{yKlnr )kIufi{kImffilKmffioKGkIiŁnŁ;LŁKnKŁnW"oK~fikiL'kImffiuoKi;r D'['
Plurisubharmonic Functions And Analytic Discs On Manifolds
, 1998
"... Let X be a complex manifold and AX be the family of maps D ! X which are holomorphic in a neighbourhood of the closure of the unit disc D . Such maps are called (closed) analytic discs in X. A disc functional on X is a map H : AX ! R[ f\Gamma1g. The envelope of H is the function EH : X ! R[ f ..."
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Cited by 34 (11 self)
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functional associated to an upper semicontinuous function ' : X ! R[ f\Gamma1g takes f 2 AX to 1 2ß R T ' ffi f d, where is the arc length measure on the unit circle T. The Riesz functional associated to a plurisubharmonic function v on X takes f to 1 2ß R D log j \Delta j \Delta(v ffi f
Subresultants Under Composition
 J. Symbolic Computation
, 1995
"... It is a well known fact that the resultants are invariant under translation. We extend this fact to arbitrary composition (where a translation is a particular composition with a linear monic polynomial), and to arbitrary subresultants (where the resultant is the 0th subresultant). 1 Introduction I ..."
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Cited by 14 (3 self)
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that their resultant remains to be 4. In general, let A(x) and B(x) be two polynomials and R be their resultant. Let A (x) = A(x + c) and B (x) = B(x + c) and let R be their resultant. Then R = R . Note that a translation (by c) is a particular composition, that is, A = A ffi C where ffi is a
About the Distance to Singularity for Polynomials in One Variable
, 1998
"... This paper is devoted to the study of ffi . In the case of a linear system Ax = b, it is known by a theorem attributed to Turing that the distance to to the closest singular system is equal, in the spectral norm k:k 2 , to 1=K(A), where K(A) = kAk 2 ..."
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This paper is devoted to the study of ffi . In the case of a linear system Ax = b, it is known by a theorem attributed to Turing that the distance to to the closest singular system is equal, in the spectral norm k:k 2 , to 1=K(A), where K(A) = kAk 2
A class of integration by parts formulae in stochastic analysis I
"... Introduction Consider a Stratonovich stochastic differential equation dx t = X(x t ) ffi dB t +A(x t )dt (1) with C 1 coefficients on a compact Riemannian manifold M , with associated differential generator A = 1 2 \Delta M +Z and solution flow f¸ t : t 0g of random smooth diffeomorphisms of ..."
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Cited by 19 (10 self)
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Introduction Consider a Stratonovich stochastic differential equation dx t = X(x t ) ffi dB t +A(x t )dt (1) with C 1 coefficients on a compact Riemannian manifold M , with associated differential generator A = 1 2 \Delta M +Z and solution flow f¸ t : t 0g of random smooth diffeomorphisms
Differentiation of Heat Semigroups and Applications
, 1994
"... Introduction Consider the Stratonovich stochastic differential equation dx t = X(x t ) ffi dB t +A(x t )dt: (1) on R n with coefficients A : R n ! R n a smooth vector field and X : R n ! L (R m ; R n ) a smooth map into the space of linear maps of R m into R n , driven by the white ..."
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Introduction Consider the Stratonovich stochastic differential equation dx t = X(x t ) ffi dB t +A(x t )dt: (1) on R n with coefficients A : R n ! R n a smooth vector field and X : R n ! L (R m ; R n ) a smooth map into the space of linear maps of R m into R n , driven
Morozov's discrepancy principle for Tikhonov regularization of severely illposed problems in finitedimensional subspaces.
"... In this paper severely illposed problems are studied which are represented in the form of linear operator equations with infinitely smoothing operators but with solutions having only a finite smoothness. It is well known, that the combination of Morozov's discrepancy principle and a finite dim ..."
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Cited by 10 (0 self)
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solution to a linear illposed problem represented in the form of an operator equation Ax = y; (1) where instead of y noisy data y ffi are available with ky \Gamma y ffi k ffi and A is a linear compact injective operator between Hilbert spaces X and Y: Usually 1 (1) is called a severely illposed problem
Concerning the Geometry of Stochastic Differential Equations and Stochastic Flows
 Kusuoka and I. Shigekawa (Eds.) New Trends in Stochastic Analysis. Proc. Taniguchi Symposium
, 1995
"... Le Jan and Watanabe showed that a nondegenerate stochastic flow f¸ t : t 0g on a manifold M determines a connection on M . This connection is characterized here and shown to be the LeviCivita connection for gradient systems. This both explains why such systems have useful properties and allows u ..."
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Cited by 16 (10 self)
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Bochner vanishing theorem. 1 Introduction and Notations A. Consider a Stratonovich stochastic differential equation dx t = X(x t ) ffi dB t +A(x t )dt (1) on an ndimensional C 1 manifold M , e.g. M = R n . Here A is a C 1 vector field on M , so A(x) lies in the tangent space T x M to M at x
Results 1  10
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28