From 291c35bc1c02fe79952f0dbfb767b02f7612d40f Mon Sep 17 00:00:00 2001 From: Samuel GOUGEON Date: Fri, 1 Jan 2021 19:30:30 +0100 Subject: [PATCH] [doc] power() improved page en_US PDF: http://bugzilla.scilab.org/attachment.cgi?id=5211 Change-Id: I31f2963d5b8967cb27215a19670df1cb503e6ce8 --- .../help/en_US/exponential/power.xml | 281 ++++++++++++------- .../help/fr_FR/exponential/power.xml | 241 ++++++++++++---- .../help/ja_JP/exponential/power.xml | 291 +++++++++++++------- .../help/pt_BR/exponential/power.xml | 106 ------- .../help/ru_RU/exponential/power.xml | 254 +++++++++++++---- 5 files changed, 742 insertions(+), 431 deletions(-) delete mode 100644 scilab/modules/elementary_functions/help/pt_BR/exponential/power.xml diff --git a/scilab/modules/elementary_functions/help/en_US/exponential/power.xml b/scilab/modules/elementary_functions/help/en_US/exponential/power.xml index 2384b7a..4c74ea6 100644 --- a/scilab/modules/elementary_functions/help/en_US/exponential/power.xml +++ b/scilab/modules/elementary_functions/help/en_US/exponential/power.xml @@ -1,138 +1,196 @@ - + + power (^,.^) power operation Syntax - t=A^b - t=A**b - t=A.^b + + t = A ^ b + t = A ** b + t = A .^ b Arguments - A,t + A, t - scalar, polynomial or rational matrix. + a scalar, vector, or matrix of encoded integers, decimal or complex numbers, + polynomials, or rationals. + b - a scalar, a vector or a scalar matrix. + a scalar, vector, or matrix of encoded integers, decimal or complex numbers. + - - - Description - - - - If A is a square matrix and b is a scalar then A^b is the matrix A to the power b. - - - - - If b is a scalar and A a matrix then - A.^b is the matrix formed by the element of - A to the power b (element-wise power). If - A is a vector and b is a scalar then - A^b and A.^b performs the same operation - (i.e. element-wise power). - - - - - If A is a scalar and b is a square matrix A^b is the matrix expm(log(A) * b). - - - If A is a scalar and b is a vector A^b and A.^b are the vector formed by a^(b(i,j)). - - - If A is a scalar and b is a matrix A.^b is the matrix formed by a^(b(i,j)). - - - - - If A and b are vectors (matrices) of the same size A.^b is the A(i)^b(i) vector (A(i,j)^b(i,j) matrix). - - - - - - Additional Remarks - - Notes: - - - 1. For square matrices A^p is computed through successive matrices - multiplications if p is a positive integer, and by diagonalization if not (see "note 2 and 3" below for details). - - - 2. If A is a square and Hermitian matrix and p is a non-integer scalar, - A^p is computed as: - - - `A^p = u*diag(diag(s).^p)*u'` (For real matrix A, only the real part of the answer is taken into account). - - - u and s are determined by `[u,s] = schur(A)` . - - - 3. If A is not a Hermitian matrix and p is a non-integer scalar, - A^p is computed as: - - - `A^p = v*diag(diag(d).^p)*inv(v)` (For real matrix A, only the real part of the answer is taken into account). - - - d and v are determined by `[d,v] = bdiag(A+0*%i)` . - - - 4. If A and p are real or complex numbers, - A^p is the principal value determined by: - - - `A^p = exp(p*log(A))` (or `A^p = exp(p*(log(abs(A))+ %i*atan(imag(A)/real(A))))` ). - - 5. If A is a square matrix and p is a real or complex number, - A.^p is the principal value computed as: + If an operand are encoded integers, the other one can be only encoded integers or real + numbers. - `A.^p = exp(p*log(A))` (same as case 4 above). - - - 6. ** and ^ operators are synonyms. - - - - Exponentiation is right-associative in Scilab contrarily to MatlabÂ® and Octave. - For example 2^3^4 is equal to 2^(3^4) in Scilab but is equal to (2^3)^4 in MatlabÂ® - and Octave. - + If A are polynomials or rationals, b can only be + a single decimal (positive or negative) integer. + Description + + .^ by-element power + + If A or b is scalar, it is first + replicated to the size of the other, with A*ones(b) or b*ones(A). + Otherwise, A and b must have the same size. + + + Then, for each element of index i, t(i) = A(i)^b(i) + is computed. + + + + ^ matricial power + + The ^ operator is equivalent to the .^ by-element power in the following cases: + + + A is scalar and b is a vector. + + + A is a vector and b is scalar. + + + Otherwise, A or b must be a scalar, + and the other one must be a square matrix: + + + + If A is scalar and b is + a square matrix, then A^b is the matrix + expm(log(A) * b) + + + + + If A is a square matrix and b + is scalar, then A^b is the matrix + A to the power b. + + + + + + + Remarks + + + + For square matrices A^p is computed through successive + matrices multiplications if p is a positive integer, and by + diagonalization if not (see "note 2 and 3" below for details). + + + + + If A is a square and Hermitian matrix and p + is a non-integer scalar, A^p is computed as: + + + `A^p = u*diag(diag(s).^p)*u'` (For real matrix A, + only the real part of the answer is taken into account). + + + u and s are determined by + `[u,s] = schur(A)` . + + + + + If A is not a Hermitian matrix and p is a + non-integer scalar, A^p is computed as: + + + `A^p = v*diag(diag(d).^p)*inv(v)` (For real matrix A, + only the real part of the answer is taken into account). + + + d and v are determined by + `[d,v] = bdiag(A+0*%i)`. + + + + + If A and p are real or complex numbers, + A^p is the principal value determined by + + + `A^p = exp(p*log(A))` + + + (or `A^p = exp(p*(log(abs(A))+ %i*atan(imag(A)/real(A))))` ). + + + + + If A is a square matrix and p is a real or + complex number, A.^p is the principal value + computed as: + + + `A.^p = exp(p*log(A))` (same as case 4 above). + + + + + ** and ^ operators are synonyms. + + + + + + Exponentiation is right-associative in Scilab, contrarily to MatlabÂ® and Octave. + For example 2^3^4 is equal to 2^(3^4) in Scilab, but to (2^3)^4 in MatlabÂ® and Octave. + + + + + Examples @@ -142,8 +200,27 @@ s^(1:10) exp + expm + + hat + + inv + + + History + + + 6.0.0 + + With decimal or complex numbers, scalar ^ squareMat now + yields expm(log(scalar)*squareMat) instead of + scalar .^ squareMat + + + + diff --git a/scilab/modules/elementary_functions/help/fr_FR/exponential/power.xml b/scilab/modules/elementary_functions/help/fr_FR/exponential/power.xml index 4872795..5760fcf 100644 --- a/scilab/modules/elementary_functions/help/fr_FR/exponential/power.xml +++ b/scilab/modules/elementary_functions/help/fr_FR/exponential/power.xml @@ -1,5 +1,21 @@ - + + power Ã©lÃ©vation Ã  la puissance (^,.^) @@ -7,87 +23,177 @@ SÃ©quence d'appel - t=A^b - t=A**b - t=A.^b + t = A ^ b + t = A ** b + t = A .^ b ParamÃ¨tres - A,t + A, t - matrice rÃ©elle, complexe, polynomiale ou rationnelle - + scalaire, vecteur, ou matrice de nombres entiers encodÃ©s, de nombres + ou polynÃ´mes ou fractions rationnelles Ã  coefficients rÃ©els ou complexes. + b - matrice rÃ©elle, complexe, polynomiale ou rationnelle - + scalaire, vecteur, ou matrice d'entiers encodÃ©s, de nombres dÃ©cimaux, + ou de nombres complexes. + - - - Description - - - - Si A est une matrice carrÃ©e et b un scalaire alors A^b est la matrice A Ã©levÃ©e Ã  la puissance b. - - - - - Si b est un un scalaire et A une matrice alors A.^b est la matrice formÃ©e par les Ã©lÃ©ments de A Ã©levÃ©s Ã  la puissance b (puissance Ã©lÃ©ment par Ã©lÃ©ment). Si A est un vecteur et b un scalaire alors A^b et A.^b donnent le mÃªme rÃ©sultat (puissance Ã©lÃ©ment par Ã©lÃ©ment). - - - - - Si A est un scalaire et b est une matrice carrÃ©e A^b est la matrice expm(log(A)*b). - - - Si A est un scalaire et b est une matrice A.^b est une matrice de mÃªme taille que b dont les termes sont Ã©gaux Ã  a^(b(i,j)). - - - - - Si A et b sont des matrices de mÃªme taille A.^b est la matrice dont les termes sont Ã©gaux Ã  A(i,j)^b(i,j). - - - - Notes : + Si un opÃ©rande sont des entiers encodÃ©s, l'autre peut Ãªtre uniquement des entiers + encodÃ©s ou des nombres rÃ©els. - - - Pour les matrices carrÃ©es A^p est calculÃ© par multiplications successives si p est un entier positif, et par diagonalisation sinon. - - - - - Les opÃ©rateurs ** et ^ sont synonymes. - - - - L'Ã©lÃ©vation Ã  la puissance est associative Ã  droite dans Scilab contrairement Ã  - MatlabÂ® et Octave. Par exemple 2^3^4 est Ã©gal Ã  2^(3^4) dans Scilab mais est Ã©gal Ã  - (2^3)^4 dans MatlabÂ® et Octave. - + Si A sont des polynÃ´mes ou des fractions rationnelles, + b peut uniquement Ãªtre un entier dÃ©cimal (positif ou nÃ©gatif). + Description + + .^ : puissances respectives par Ã©lÃ©ment + + Si A ou b est scalaire, il est prÃ©alablement + rÃ©pliquÃ© Ã  la taille de l'autre, par A*ones(b) ou b*ones(A). + Sinon, A et b doivent avoir la mÃªme taille. + + + Alors, pour chaque Ã©lÃ©ment numÃ©ro i, t(i) = A(i)^b(i) + est calculÃ©. + + + + ^ : puissance matricielle + + L'opÃ©rateur ^ est Ã©quivalent Ã  .^ dans les cas suivants : + + + A est scalaire et b est un vecteur. + + + A est un vecteur et b est scalaire. + + + Sinon, A ou b doit Ãªtre scalaire, et l'autre + opÃ©rande doit Ãªtre une matrice carrÃ©e : + + + + Si A est scalaire et b est carrÃ©e, + alors A^b est la matrice + expm(log(A) * b) + + + + + Si A est carrÃ©e et b est scalaire, + alors A^b est la matrice + A Ã  la puissane b. + + + + + + + Autres remarques + + + + Si A est une matrice carrÃ©e, A^p est + calculÃ© par multiplications successives si p est un + entier positif, et par diagonalisation sinon (dÃ©tails en remarques nÂ°2 et 3 + ci-dessous). + + + + + If A is a square and Hermitian matrix and p + is a non-integer scalar, A^p is computed as: + + + `A^p = u*diag(diag(s).^p)*u'` (For real matrix A, + only the real part of the answer is taken into account). + + + u and s are determined by + `[u,s] = schur(A)` . + + + + + If A is not a Hermitian matrix and p is a + non-integer scalar, A^p is computed as: + + + `A^p = v*diag(diag(d).^p)*inv(v)` (For real matrix A, + only the real part of the answer is taken into account). + + + d and v are determined by + `[d,v] = bdiag(A+0*%i)`. + + + + + If A and p are real or complex numbers, + A^p is the principal value determined by + + + `A^p = exp(p*log(A))` + + + (or `A^p = exp(p*(log(abs(A))+ %i*atan(imag(A)/real(A))))` ). + + + + + If A is a square matrix and p is a real or + complex number, A.^p is the principal value + computed as: + + + `A.^p = exp(p*log(A))` (same as case 4 above). + + + + + Les opÃ©rateurs ** et ^ sont Ã©quivalents. + + + + + + L'Ã©lÃ©vation Ã  la puissance est associative Ã  droite dans Scilab contrairement Ã  + MatlabÂ® et Octave. Par exemple 2^3^4 est Ã©gal Ã  2^(3^4) dans Scilab mais est Ã©gal Ã  + (2^3)^4 dans MatlabÂ® et Octave. + + + + + Exemples @@ -97,8 +203,27 @@ s^(1:10) exp + expm + + hat + + inv + + + Historique + + + 6.0.0 + + Avec des nombres dÃ©cimaux ou complexes, scalaire ^ matriceCarrÃ©e + produit dÃ©sormais expm(log(scalaire)*matriceCarrÃ©e) au lieu + de scalaire .^ matriceCarrÃ©e. + + + + diff --git a/scilab/modules/elementary_functions/help/ja_JP/exponential/power.xml b/scilab/modules/elementary_functions/help/ja_JP/exponential/power.xml index 28ad413..bab3fc1 100644 --- a/scilab/modules/elementary_functions/help/ja_JP/exponential/power.xml +++ b/scilab/modules/elementary_functions/help/ja_JP/exponential/power.xml @@ -1,142 +1,202 @@ - + + power ææ°æ¼ç®å­ (^,.^) å¼åºãæé  - t=A^b - t=A**b - t=A.^b + + t = A ^ b + t = A ** b + t = A .^ b å¼æ° - A,t + A, t - ã¹ã«ã©ã¼, å¤é å¼ã¾ãã¯æçè¡å. + a scalar, vector, or matrix of encoded integers, decimal or complex numbers, + polynomials, or rationals. + b - ã¹ã«ã©ã¼, ãã¯ãã«ã¾ãã¯ã¹ã«ã©ã¼ã®è¡å. + a scalar, vector, or matrix of encoded integers, decimal or complex numbers. + - - - èª¬æ - - - - A ãæ­£æ¹è¡åã§b ãã¹ã«ã©ã¼ã®å ´å, - A^bã¯è¡åAã®bä¹ã« - ãªãã¾ã. - - - - - b ãã¹ã«ã©ã¼ã§Aãè¡åã®å ´å, - A.^bã¯Aã®åè¦ç´ ãbä¹ - (è¦ç´ æ¯ã®ç´¯ä¹)ã«ããè¡åã¨ãªãã¾ã. - A ããã¯ãã«ã§ b ãã¹ã«ã©ã¼ã®å ´å, - A^b ã¨ A.^b ã¯åãæå³ã¨ãªãã¾ã - (ããªãã¡,è¦ç´ æ¯ã®ç´¯ä¹). - - - - - A ãã¹ã«ã©ã¼ã§,b ãè¡å (ã¾ãã¯ãã¯ãã«)ã®å ´å, - A^b ããã³ A.^b ã¯, - a^(b(i,j)) ã«ããæ§æãããè¡å (ã¾ãã¯ãã¯ãã«) ã¨ãªãã¾ã. - - - - - A ããã³ b ãåãå¤§ããã®ãã¯ãã« (è¡å) ã®å ´å, - A.^b ã¯ãã¯ãã« A(i)^b(i) - (è¡åA(i,j)^b(i,j))ã¨ãªãã¾ã. - - - - - - è¿½å ã®æ³¨è¨ - - æ³¨æ: - - - 1.æ­£æ¹è¡åã®å ´å, A^pã¯, - pãæ­£ã®ã¹ã«ã©ã¼ã®å ´åã¯è¡åã®éæ¬¡ä¹ç®ã«ããè¨ç®ãã, - ããä»¥å¤ã®å ´å,å¯¾è§åã«ããè¨ç®ããã¾ã - (è©³ç´°ã¯"æ³¨è¨2ããã³3"ãåç§). - - - 2. Aãæ­£æ¹ãã¤ã¨ã«ãã¼ãè¡åã§ - p ãæ´æ°ã§ãªãã¹ã«ã©ã¼ã®å ´å, - A^p ã¯ä»¥ä¸ã®æ§ã«è¨ç®ããã¾ã: - - - `A^p = u*diag(diag(s).^p)*u'` (Aãå®æ°è¡åã®å ´å, - ç­ãã®å®é¨ã®ã¿ãèæ®ããã¾ã). - - - uããã³s ã¯, `[u,s] = schur(A)` - ã«ããå®ç¾©ããã¾ã. - - - 3. A ãã¨ã«ãã¼ãè¡åã§ãªã, - p ãéæ´æ°ã¹ã«ã©ã¼ã®å ´å, - A^p ã¯ä»¥ä¸ã®æ§ã«è¨ç®ããã¾ã: - - - `A^p = v*diag(diag(d).^p)*inv(v)` - (Aãå®æ°è¡åã®å ´å, ç­ãã®å®é¨ã®ã¿ãèæ®ããã¾ã). - - - d ããã³ v ã¯, - `[d,v] = bdiag(A+0*%i)`ã«ããå®ç¾©ããã¾ã. - - 4. A ããã³ p ãå®æ°ã¾ãã¯è¤ç´ æ°ã®å ´å, - A^p ã¯ä»¥ä¸ã®ããã«è¨ç®ããã - ä¸»å¤ã¨ãªãã¾ã: + If an operand are encoded integers, the other one can be only encoded integers or real + numbers. - `A^p = exp(p*log(A))` (ã¾ãã¯`A^p = exp(p*(log(abs(A))+ %i*atan(imag(A)/real(A))))` ). - - - 5. A ãæ­£æ¹è¡åã§ - p ãå®æ°ã¾ãã¯è¤ç´ æ°ã®å ´å, - A.^p ã¯ä»¥ä¸ã®ããã«è¨ç®ããã - ä¸»å¤ ã¨ãªãã¾ã: - - - `A.^p = exp(p*log(A))` (ä¸è¨ã®ã±ã¼ã¹4ã¨åã). - - - 6. ** ããã³ ^ æ¼ç®å­ã¯åç¾©ã§ã. + If A are polynomials or rationals, b can only be + a single decimal (positive or negative) integer. + èª¬æ + + .^ by-element power + + If A or b is scalar, it is first + replicated to the size of the other, with A*ones(b) or b*ones(A). + Otherwise, A and b must have the same size. + + + Then, for each element of index i, t(i) = A(i)^b(i) + is computed. + + + + ^ matricial power + + The ^ operator is equivalent to the .^ by-element power in the following cases: + + + A is scalar and b is a vector. + + + A is a vector and b is scalar. + + + Otherwise, A or b must be a scalar, + and the other one must be a square matrix: + + + + If A is scalar and b is + a square matrix, then A^b is the matrix + expm(log(A) * b) + + + + + If A is a square matrix and b + is scalar, then A^b is the matrix + A to the power b. + + + + + + + è¿½å ã®æ³¨è¨ + + + + æ­£æ¹è¡åã®å ´å, A^pã¯, + pãæ­£ã®ã¹ã«ã©ã¼ã®å ´åã¯è¡åã®éæ¬¡ä¹ç®ã«ããè¨ç®ãã, + ããä»¥å¤ã®å ´å,å¯¾è§åã«ããè¨ç®ããã¾ã + (è©³ç´°ã¯"æ³¨è¨2ããã³3"ãåç§). + + + + + Aãæ­£æ¹ãã¤ã¨ã«ãã¼ãè¡åã§ + p ãæ´æ°ã§ãªãã¹ã«ã©ã¼ã®å ´å, + A^p ã¯ä»¥ä¸ã®æ§ã«è¨ç®ããã¾ã: + + + `A^p = u*diag(diag(s).^p)*u'` (Aãå®æ°è¡åã®å ´å, + ç­ãã®å®é¨ã®ã¿ãèæ®ããã¾ã). + + + uããã³s ã¯, `[u,s] = schur(A)` + ã«ããå®ç¾©ããã¾ã. + + + + + A ãã¨ã«ãã¼ãè¡åã§ãªã, + p ãéæ´æ°ã¹ã«ã©ã¼ã®å ´å, + A^p ã¯ä»¥ä¸ã®æ§ã«è¨ç®ããã¾ã: + + + `A^p = v*diag(diag(d).^p)*inv(v)` + (Aãå®æ°è¡åã®å ´å, ç­ãã®å®é¨ã®ã¿ãèæ®ããã¾ã). + + + d ããã³ v ã¯, + `[d,v] = bdiag(A+0*%i)`ã«ããå®ç¾©ããã¾ã. + + + + + A ããã³ p ãå®æ°ã¾ãã¯è¤ç´ æ°ã®å ´å, + A^p ã¯ä»¥ä¸ã®ããã«è¨ç®ããã + ä¸»å¤ã¨ãªãã¾ã: + + + `A^p = exp(p*log(A))` + + + (ã¾ãã¯`A^p = exp(p*(log(abs(A))+ %i*atan(imag(A)/real(A))))` ). + + + + + A ãæ­£æ¹è¡åã§ + p ãå®æ°ã¾ãã¯è¤ç´ æ°ã®å ´å, + A.^p ã¯ä»¥ä¸ã®ããã«è¨ç®ããã + ä¸»å¤ ã¨ãªãã¾ã: + + + `A.^p = exp(p*log(A))` (ä¸è¨ã®ã±ã¼ã¹4ã¨åã). + + + + + ** ããã³ ^ æ¼ç®å­ã¯åç¾©ã§ã. + + + + + + Exponentiation is right-associative in Scilab, contrarily to MatlabÂ® and Octave. + For example 2^3^4 is equal to 2^(3^4) in Scilab, but to (2^3)^4 in MatlabÂ® and + Octave. + + + + + ä¾ @@ -146,8 +206,27 @@ s^(1:10) exp + expm + + hat + + inv + + + å±¥æ­´ + + + 6.0.0 + + With decimal or complex numbers, scalar ^ squareMat now + yields expm(log(scalar)*squareMat) instead of + scalar .^ squareMat + + + + diff --git a/scilab/modules/elementary_functions/help/pt_BR/exponential/power.xml b/scilab/modules/elementary_functions/help/pt_BR/exponential/power.xml deleted file mode 100644 index 8857645..0000000 --- a/scilab/modules/elementary_functions/help/pt_BR/exponential/power.xml +++ /dev/null @@ -1,106 +0,0 @@ - - - - power - operaÃ§Ã£o de potenciaÃ§Ã£o(^,.^) - - - SeqÃ¼Ãªncia de Chamamento - t=A^b - t=A**b - t=A.^b - - - - ParÃ¢metros - - - A,t - - matriz de escalares, polinÃ´mios ou razÃµes de - polinÃ´mios. - - - - - b - - um escalar ou um vetor ou matriz de escalares. - - - - - - DescriÃ§Ã£o - - - - "(A:square)^(b:scalar)"Se - A Ã© uma matriz quadrada e b Ã© um - escalar, entÃ£o A^b Ã© a matriz A - elevada Ã  potÃªncia b. - - - - - "(A:matrix).^(b:scalar)"Se - b Ã© um escalar e A uma matriz, - entÃ£o A.^b Ã© formada pelos elementos de - A elevados Ã  potÃªncia b - (potenciaÃ§Ã£o elemento a elemento). Se A Ã© um vetor - e b Ã© um escalar, entÃ£o A^b e - A.^b realizam a mesma operaÃ§Ã£o (i.e., potenciaÃ§Ã£o - elemento a elemento). - - - - - "(A:scalar).^(b:matrix)" Se - A Ã© um escalar e b Ã© uma matriz - (ou vetor) entÃ£o A^b e A.^b sÃ£o - as matrizes (ou vetores) formados por - a^(b(i,j)) - - . - - - - - "(A:matrix).^(b:matrix)" Se - A e b sÃ£o vetores (matrizes) de - mesmo tamanho A.^b Ã© o vetor - A(i)^b(i) (matriz - A(i,j)^b(i,j)). - - - - Notas: - - - Para matrizes quadradas A^p Ã© computada atravÃ©s - de sucessivas multiplicaÃ§Ãµes de matrizes se p is Ã© um - nÃºmero inteiro positivo e por diagonalizaÃ§Ã£o se nÃ£o for. - - - Os operadores ** e ^ sÃ£o sinÃ´nimos. - - - Exemplos - - - - Ver TambÃ©m - - - exp - - - - diff --git a/scilab/modules/elementary_functions/help/ru_RU/exponential/power.xml b/scilab/modules/elementary_functions/help/ru_RU/exponential/power.xml index e62bb7c..1c30650 100644 --- a/scilab/modules/elementary_functions/help/ru_RU/exponential/power.xml +++ b/scilab/modules/elementary_functions/help/ru_RU/exponential/power.xml @@ -1,5 +1,21 @@ - + + Ð²Ð¾Ð·Ð²ÐµÐ´ÐµÐ½Ð¸Ðµ Ð² ÑÑÐµÐ¿ÐµÐ½Ñ Ð¾Ð¿ÐµÑÐ°ÑÐ¸Ñ Ð²Ð¾Ð·Ð²ÐµÐ´ÐµÐ½Ð¸Ñ Ð² ÑÑÐµÐ¿ÐµÐ½Ñ (^, .^) @@ -7,88 +23,189 @@ Ð¡Ð¸Ð½ÑÐ°ÐºÑÐ¸Ñ - t=A^b - t=A**b - t=A.^b + t = A ^ b + t = A ** b + t = A .^ b ÐÑÐ³ÑÐ¼ÐµÐ½ÑÑ - A,t + A, t - - ÑÐºÐ°Ð»ÑÑ Ð¸Ð»Ð¸ Ð²ÐµÐºÑÐ¾Ñ/Ð¼Ð°ÑÑÐ¸ÑÐ° ÑÐ¸ÑÐ»Ð¾Ð²ÑÑ, Ð¿Ð¾Ð»Ð¸Ð½Ð¾Ð¼Ð¸Ð°Ð»ÑÐ½ÑÑ Ð¸Ð»Ð¸ ÑÐ°ÑÐ¸Ð¾Ð½Ð°Ð»ÑÐ½ÑÑ Ð·Ð½Ð°ÑÐµÐ½Ð¸Ð¹ - + ÑÐºÐ°Ð»ÑÑ, Ð²ÐµÐºÑÐ¾Ñ Ð¸Ð»Ð¸ Ð¼Ð°ÑÑÐ¸ÑÐ° ÐºÐ¾Ð´Ð¸ÑÐ¾Ð²Ð°Ð½Ð½ÑÑ ÑÐµÐ»ÑÑ ÑÐ¸ÑÐµÐ», Ð´ÐµÑÑÑÐ¸ÑÐ½ÑÑ Ð¸Ð»Ð¸ + ÐºÐ¾Ð¼Ð¿Ð»ÐµÐºÑÐ½ÑÑ ÑÐ¸ÑÐµÐ», Ð¿Ð¾Ð»Ð¸Ð½Ð¾Ð¼Ð¾Ð² Ð¸Ð»Ð¸ Ð´ÑÐ¾Ð±Ð½Ð¾-ÑÐ°ÑÐ¸Ð¾Ð½Ð°Ð»ÑÐ½ÑÑ Ð²ÑÑÐ°Ð¶ÐµÐ½Ð¸Ð¹. + b - ÑÐºÐ°Ð»ÑÑ, Ð²ÐµÐºÑÐ¾Ñ Ð¸Ð»Ð¸ Ð¼Ð°ÑÑÐ¸ÑÐ°. + ÑÐºÐ°Ð»ÑÑ, Ð²ÐµÐºÑÐ¾Ñ Ð¸Ð»Ð¸ Ð¼Ð°ÑÑÐ¸ÑÐ° ÐºÐ¾Ð´Ð¸ÑÐ¾Ð²Ð°Ð½Ð½ÑÑ ÑÐµÐ»ÑÑ ÑÐ¸ÑÐµÐ», Ð´ÐµÑÑÑÐ¸ÑÐ½ÑÑ Ð¸Ð»Ð¸ + ÐºÐ¾Ð¼Ð¿Ð»ÐµÐºÑÐ½ÑÑ ÑÐ¸ÑÐµÐ». + - - - ÐÐ¿Ð¸ÑÐ°Ð½Ð¸Ðµ - - - - ÐÑÐ»Ð¸ A -- ÐºÐ²Ð°Ð´ÑÐ°ÑÐ½Ð°Ñ Ð¼Ð°ÑÑÐ¸ÑÐ°, Ð° b -- ÑÐºÐ°Ð»ÑÑ, ÑÐ¾ A^b - ÑÐ²Ð»ÑÐµÑÑÑ Ð¼Ð°ÑÑÐ¸ÑÐµÐ¹ A Ð² ÑÑÐµÐ¿ÐµÐ½Ð¸ b. - - - - - ÐÑÐ»Ð¸ A -- Ð¼Ð°ÑÑÐ¸ÑÐ°, Ð° b -- ÑÐºÐ°Ð»ÑÑ, ÑÐ¾ Ð¼Ð°ÑÑÐ¸ÑÐ° - A.^b ÑÐ¾ÑÐ¼Ð¸ÑÑÐµÑÑÑ ÑÐ»ÐµÐ¼ÐµÐ½ÑÐ°Ð¼Ð¸ Ð¼Ð°ÑÑÐ¸ÑÑ A - Ð² ÑÑÐµÐ¿ÐµÐ½Ð¸ b (Ð¿Ð¾ÑÐ»ÐµÐ¼ÐµÐ½ÑÐ½Ð¾Ðµ Ð²Ð¾Ð·Ð²ÐµÐ´ÐµÐ½Ð¸Ðµ Ð² ÑÑÐµÐ¿ÐµÐ½Ñ). - ÐÑÐ»Ð¸ A -- Ð²ÐµÐºÑÐ¾Ñ, Ð° b -- ÑÐºÐ°Ð»ÑÑ, ÑÐ¾ - A^b Ð¸ A.^b Ð²ÑÐ¿Ð¾Ð»Ð½ÑÑÑ Ð¾Ð´Ð½Ñ Ð¸ ÑÑ Ð¶Ðµ Ð¾Ð¿ÐµÑÐ°ÑÐ¸Ñ - (Ñ. Ðµ. Ð¿Ð¾ÑÐ»ÐµÐ¼ÐµÐ½ÑÐ½Ð¾Ðµ Ð²Ð¾Ð·Ð²ÐµÐ´ÐµÐ½Ð¸Ðµ Ð² ÑÑÐµÐ¿ÐµÐ½Ñ). - - - - - ÐÑÐ»Ð¸ A -- ÑÐºÐ°Ð»ÑÑ, Ð° b -- Ð¼Ð°ÑÑÐ¸ÑÐ° (Ð¸Ð»Ð¸ Ð²ÐµÐºÑÐ¾Ñ), ÑÐ¾ - A^b Ð¸ A.^b ÑÐ²Ð»ÑÑÑÑÑ Ð¼Ð°ÑÑÐ¸ÑÐ°Ð¼Ð¸ (Ð¸Ð»Ð¸ Ð²ÐµÐºÑÐ¾ÑÐ°Ð¼Ð¸), ÑÑÐ¾ÑÐ¼Ð¸ÑÐ¾Ð²Ð°Ð½Ð½ÑÐ¼Ð¸ - ÑÐ»ÐµÐ¼ÐµÐ½ÑÐ°Ð¼Ð¸ a^(b(i,j)). - - - - - ÐÑÐ»Ð¸ A Ð¸ b -- Ð²ÐµÐºÑÐ¾ÑÑ (Ð¼Ð°ÑÑÐ¸ÑÑ) Ð¾Ð´Ð½Ð¾Ð³Ð¾ ÑÐ°Ð·Ð¼ÐµÑÐ°, ÑÐ¾ - A.^b ÑÐ°Ð²Ð½Ð¾ A(i)^b(i) (Ð²ÐµÐºÑÐ¾ÑÑ) Ð¸Ð»Ð¸ A(i,j)^b(i,j) (Ð¼Ð°ÑÑÐ¸ÑÑ). - - - - - ÐÑÐ¸Ð¼ÐµÑÐ°Ð½Ð¸Ñ: - - - - ÐÐ»Ñ ÐºÐ²Ð°Ð´ÑÐ°ÑÐ½ÑÑ Ð¼Ð°ÑÑÐ¸Ñ A^p Ð²ÑÑÐ¸ÑÐ»ÑÐµÑÑÑ ÑÐµÑÐµÐ· Ð¿Ð¾ÑÐ»ÐµÐ´Ð¾Ð²Ð°ÑÐµÐ»ÑÐ½Ð¾Ðµ - Ð¿ÐµÑÐµÐ¼Ð½Ð¾Ð¶ÐµÐ½Ð¸Ðµ Ð¼Ð°ÑÑÐ¸Ñ, ÐµÑÐ»Ð¸ p ÑÐ²Ð»ÑÐµÑÑÑ Ð¿Ð¾Ð»Ð¾Ð¶Ð¸ÑÐµÐ»ÑÐ½ÑÐ¼ ÑÐ¸ÑÐ»Ð¾Ð¼, Ð° Ð¸Ð½Ð°ÑÐµ -- - ÑÐµÑÐµÐ· Ð´Ð¸Ð°Ð³Ð¾Ð½Ð°Ð»Ð¸Ð·Ð°ÑÐ¸Ñ. + ÐÑÐ»Ð¸ Ð¾Ð¿ÐµÑÐ°Ð½Ð´Ð¾Ð¼ ÑÐ²Ð»ÑÑÑÑÑ ÐºÐ¾Ð´Ð¸ÑÐ¾Ð²Ð°Ð½Ð½ÑÐµ ÑÐµÐ»ÑÐµ ÑÐ¸ÑÐ»Ð°, ÑÐ¾ Ð´ÑÑÐ³Ð¸Ðµ ÑÐ¸ÑÐ»Ð° Ð¼Ð¾Ð³ÑÑ + Ð±ÑÑÑ ÑÐ¾Ð»ÑÐºÐ¾ ÐºÐ¾Ð´Ð¸ÑÐ¾Ð²Ð°Ð½Ð½ÑÐ¼Ð¸ ÑÐµÐ»ÑÐ¼Ð¸ ÑÐ¸ÑÐ»Ð°Ð¼Ð¸ Ð¸Ð»Ð¸ Ð²ÐµÑÐµÑÑÐ²ÐµÐ½Ð½ÑÐ¼Ð¸ ÑÐ¸ÑÐ»Ð°Ð¼Ð¸. - - - Ð¾Ð¿ÐµÑÐ°ÑÐ¾ÑÑ ** Ð¸ ^ ÑÐ²Ð»ÑÑÑÑÑ ÑÐ¸Ð½Ð¾Ð½Ð¸Ð¼Ð°Ð¼Ð¸. + ÐÑÐ»Ð¸ A ÑÐ²Ð»ÑÐµÑÑÑ Ð¿Ð¾Ð»Ð¸Ð½Ð¾Ð¼Ð°Ð¼Ð¸ Ð¸Ð»Ð¸ Ð´ÑÐ¾Ð±Ð½Ð¾-ÑÐ°ÑÐ¸Ð¾Ð½Ð°Ð»ÑÐ½ÑÐ¼Ð¸ + Ð²ÑÑÐ°Ð¶ÐµÐ½Ð¸ÑÐ¼Ð¸, ÑÐ¾ b Ð¼Ð¾Ð¶ÐµÑ Ð±ÑÑÑ ÑÐ¾Ð»ÑÐºÐ¾ Ð¾Ð´Ð¸Ð½Ð¾ÑÐ½ÑÐ¼ Ð´ÐµÑÑÑÐ¸ÑÐ½ÑÐ¼ + (Ð¿Ð¾Ð»Ð¾Ð¶Ð¸ÑÐµÐ»ÑÐ½ÑÐ¼ Ð¸Ð»Ð¸ Ð¾ÑÑÐ¸ÑÐ°ÑÐµÐ»ÑÐ½ÑÐ¼) ÑÐ¸ÑÐ»Ð¾Ð¼. + ÐÐ¿Ð¸ÑÐ°Ð½Ð¸Ðµ + + .^ Ð¿Ð¾ÑÐ»ÐµÐ¼ÐµÐ½ÑÐ½Ð¾Ðµ Ð²Ð¾Ð·Ð²ÐµÐ´ÐµÐ½Ð¸Ðµ Ð² ÑÑÐµÐ¿ÐµÐ½Ñ + + ÐÑÐ»Ð¸ A Ð¸Ð»Ð¸ b ÑÐºÐ°Ð»ÑÑ, + ÑÐ¾ Ð¾Ð½ ÑÐ½Ð°ÑÐ°Ð»Ð° ÑÐµÐ¿Ð»Ð¸ÑÐ¸ÑÑÐµÑÑÑ Ð´Ð¾ ÑÐ°Ð·Ð¼ÐµÑÐ° Ð´ÑÑÐ³Ð¾Ð³Ð¾ Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ + A*ones(b) Ð¸Ð»Ð¸ b*ones(A). + Ð Ð¿ÑÐ¾ÑÐ¸Ð²Ð½Ð¾Ð¼ ÑÐ»ÑÑÐ°Ðµ A Ð¸ b + Ð´Ð¾Ð»Ð¶Ð½Ñ Ð±ÑÑÑ Ð¾Ð´Ð¸Ð½Ð°ÐºÐ¾Ð²Ð¾Ð³Ð¾ ÑÐ°Ð·Ð¼ÐµÑÐ°. + + + ÐÐ°ÑÐµÐ¼ Ð´Ð»Ñ ÐºÐ°Ð¶Ð´Ð¾Ð³Ð¾ ÑÐ»ÐµÐ¼ÐµÐ½ÑÐ° Ñ Ð¸Ð½Ð´ÐµÐºÑÐ¾Ð¼ i + Ð²ÑÑÐ¸ÑÐ»ÑÐµÑÑÑ t(i) = A(i)^b(i). + + + + ^ Ð¼Ð°ÑÑÐ¸ÑÐ½Ð¾Ðµ Ð²Ð¾Ð·Ð²ÐµÐ´ÐµÐ½Ð¸Ðµ Ð² ÑÑÐµÐ¿ÐµÐ½Ñ + + ÐÐ¿ÐµÑÐ°ÑÐ¾Ñ ^ ÑÐºÐ²Ð¸Ð²Ð°Ð»ÐµÐ½ÑÐµÐ½ Ð¿Ð¾ÑÐ»ÐµÐ¼ÐµÐ½ÑÐ½Ð¾Ð¼Ñ Ð²Ð¾Ð·Ð²ÐµÐ´ÐµÐ½Ð¸Ñ + Ð² ÑÑÐµÐ¿ÐµÐ½Ñ .^ Ð² ÑÐ»ÐµÐ´ÑÑÑÐ¸Ñ ÑÐ»ÑÑÐ°ÑÑ: + + + A ÑÐºÐ°Ð»ÑÑ, Ð° b Ð²ÐµÐºÑÐ¾Ñ; + + + A Ð²ÐµÐºÑÐ¾Ñ, Ð° b ÑÐºÐ°Ð»ÑÑ. + + + Ð Ð¿ÑÐ¾ÑÐ¸Ð²Ð½Ð¾Ð¼ ÑÐ»ÑÑÐ°Ðµ A Ð»Ð¸Ð±Ð¾ b + Ð´Ð¾Ð»Ð¶ÐµÐ½ Ð±ÑÑÑ ÑÐºÐ°Ð»ÑÑÐ¾Ð¼, Ð° Ð´ÑÑÐ³Ð¾Ð¹ Ð´Ð¾Ð»Ð¶ÐµÐ½ Ð±ÑÑÑ ÐºÐ²Ð°Ð´ÑÐ°ÑÐ½Ð¾Ð¹ Ð¼Ð°ÑÑÐ¸ÑÐµÐ¹: + + + + ÐµÑÐ»Ð¸ A ÑÐºÐ°Ð»ÑÑ, Ð° b + ÐºÐ²Ð°Ð´ÑÐ°ÑÐ½Ð°Ñ Ð¼Ð°ÑÑÐ¸ÑÐ°, ÑÐ¾ A^b ÑÐ²Ð»ÑÐµÑÑÑ + Ð¼Ð°ÑÑÐ¸ÑÐµÐ¹ expm(log(A) * b); + + + + + ÐµÑÐ»Ð¸ A ÐºÐ²Ð°Ð´ÑÐ°ÑÐ½Ð°Ñ Ð¼Ð°ÑÑÐ¸ÑÐ°, Ð° b + ÑÐºÐ°Ð»ÑÑ, ÑÐ¾ A^b ÑÐ²Ð»ÑÐµÑÑÑ Ð¼Ð°ÑÑÐ¸ÑÐµÐ¹ + A Ð² ÑÑÐµÐ¿ÐµÐ½Ð¸ b. + + + + + + + ÐÑÐ¸Ð¼ÐµÑÐ°Ð½Ð¸Ñ + + + + ÐÐ»Ñ ÐºÐ²Ð°Ð´ÑÐ°ÑÐ½ÑÑ Ð¼Ð°ÑÑÐ¸Ñ A, A^p + Ð²ÑÑÐ¸ÑÐ»ÑÐµÑÑÑ ÑÐµÑÐµÐ· Ð¿Ð¾ÑÐ»ÐµÐ´Ð¾Ð²Ð°ÑÐµÐ»ÑÐ½Ð¾Ðµ Ð¿ÐµÑÐµÐ¼Ð½Ð¾Ð¶ÐµÐ½Ð¸Ðµ Ð¼Ð°ÑÑÐ¸Ñ, ÐµÑÐ»Ð¸ + p ÑÐ²Ð»ÑÐµÑÑÑ Ð¿Ð¾Ð»Ð¾Ð¶Ð¸ÑÐµÐ»ÑÐ½ÑÐ¼ ÑÐ¸ÑÐ»Ð¾Ð¼, Ð° Ð¸Ð½Ð°ÑÐµ - + ÑÐµÑÐµÐ· Ð´Ð¸Ð°Ð³Ð¾Ð½Ð°Ð»Ð¸Ð·Ð°ÑÐ¸Ñ (ÑÐ¼. Ð¿ÑÐ¸Ð¼ÐµÑÐ°Ð½Ð¸Ñ â2 Ð¸ â3 Ð½Ð¸Ð¶Ðµ). + + + + + ÐÑÐ»Ð¸ A ÐºÐ²Ð°Ð´ÑÐ°ÑÐ½Ð°Ñ Ð¸ ÑÑÐ¼Ð¸ÑÐ¾Ð²Ð° Ð¼Ð°ÑÑÐ¸ÑÐ°, Ð° + p Ð½ÐµÑÐµÐ»ÑÐ¹ ÑÐºÐ°Ð»ÑÑ, ÑÐ¾ A^p + Ð²ÑÑÐ¸ÑÐ»ÑÐµÑÑÑ ÐºÐ°Ðº: + + + `A^p = u*diag(diag(s).^p)*u'` (Ð´Ð»Ñ Ð²ÐµÑÐµÑÑÐ²ÐµÐ½Ð½Ð¾Ð¹ + Ð¼Ð°ÑÑÐ¸ÑÑ A Ð²Ð¾ Ð²Ð½Ð¸Ð¼Ð°Ð½Ð¸Ðµ Ð¿ÑÐ¸Ð½Ð¸Ð¼Ð°ÐµÑÑÑ ÑÐ¾Ð»ÑÐºÐ¾ + Ð²ÐµÑÐµÑÑÐ²ÐµÐ½Ð½Ð°Ñ ÑÐ°ÑÑÑ Ð¾ÑÐ²ÐµÑÐ°). + + + u Ð¸ s Ð¾Ð¿ÑÐµÐ´ÐµÐ»ÑÑÑÑÑ ÐºÐ°Ðº + `[u,s] = schur(A)` . + + + + + ÐÑÐ»Ð¸ A Ð½Ðµ ÑÐ²Ð»ÑÐµÑÑÑ ÑÑÐ¼Ð¸ÑÐ¾Ð²Ð¾Ð¹ Ð¼Ð°ÑÑÐ¸ÑÐµÐ¹, + Ð° p ÑÐ²Ð»ÑÐµÑÑÑ Ð½ÐµÑÐµÐ»ÑÐ¼ ÑÐºÐ°Ð»ÑÑÐ¾Ð¼, ÑÐ¾ + A^p Ð²ÑÑÐ¸ÑÐ»ÑÐµÑÑÑ ÐºÐ°Ðº: + + + `A^p = v*diag(diag(d).^p)*inv(v)` (Ð´Ð»Ñ Ð²ÐµÑÐµÑÑÐ²ÐµÐ½Ð½Ð¾Ð¹ + Ð¼Ð°ÑÑÐ¸ÑÑ A Ð²Ð¾ Ð²Ð½Ð¸Ð¼Ð°Ð½Ð¸Ðµ Ð¿ÑÐ¸Ð½Ð¸Ð¼Ð°ÐµÑÑÑ ÑÐ¾Ð»ÑÐºÐ¾ + Ð²ÐµÑÐµÑÑÐ²ÐµÐ½Ð½Ð°Ñ ÑÐ°ÑÑÑ Ð¾ÑÐ²ÐµÑÐ°). + + + d Ð¸ v Ð¾Ð¿ÑÐµÐ´ÐµÐ»ÑÑÑÑÑ + ÐºÐ°Ðº `[d,v] = bdiag(A+0*%i)`. + + + + + ÐÑÐ»Ð¸ A Ð¸ p Ð²ÐµÑÐµÑÑÐ²ÐµÐ½Ð½ÑÐµ + Ð¸Ð»Ð¸ ÐºÐ¾Ð¼Ð¿Ð»ÐµÐºÑÐ½ÑÐµ ÑÐ¸ÑÐ»Ð°, ÑÐ¾ A^p ÑÐ²Ð»ÑÐµÑÑÑ + Ð³Ð»Ð°Ð²Ð½ÑÐ¼ Ð·Ð½Ð°ÑÐµÐ½Ð¸ÐµÐ¼, Ð¾Ð¿ÑÐµÐ´ÐµÐ»ÑÐµÐ¼ÑÐ¼ ÐºÐ°Ðº + + + `A^p = exp(p*log(A))` + + + (Ð¸Ð»Ð¸ `A^p = exp(p*(log(abs(A))+ %i*atan(imag(A)/real(A))))` ). + + + + + ÐÑÐ»Ð¸ A ÑÐ²Ð»ÑÐµÑÑÑ ÐºÐ²Ð°Ð´ÑÐ°ÑÐ½Ð¾Ð¹ Ð¼Ð°ÑÑÐ¸ÑÐµ, Ð° + p Ð²ÐµÑÐµÑÑÐ²ÐµÐ½Ð½ÑÐ¼ Ð¸Ð»Ð¸ ÐºÐ¾Ð¼Ð¿Ð»ÐµÐºÑÐ½ÑÐ¼ ÑÐ¸ÑÐ»Ð¾Ð¼, ÑÐ¾ + A.^p ÑÐ²Ð»ÑÐµÑÑÑ Ð³Ð»Ð°Ð²Ð½ÑÐ¼ Ð·Ð½Ð°ÑÐµÐ½Ð¸ÐµÐ¼ + Ð²ÑÑÐ¸ÑÐ»ÐµÐ½Ð½ÑÐ¼ ÐºÐ°Ðº: + + + `A.^p = exp(p*log(A))` (ÑÐ¾ Ð¶Ðµ ÑÐ°Ð¼Ð¾Ðµ, ÑÑÐ¾ Ð¸ Ð² ÑÐ»ÑÑÐ°Ðµ 4 Ð²ÑÑÐµ). + + + + + Ð¾Ð¿ÐµÑÐ°ÑÐ¾ÑÑ ** Ð¸ ^ ÑÐ²Ð»ÑÑÑÑÑ + ÑÐ¸Ð½Ð¾Ð½Ð¸Ð¼Ð°Ð¼Ð¸. + + + + + + ÐÐ¾Ð·Ð²ÐµÐ´ÐµÐ½Ð¸Ðµ Ð² ÑÑÐµÐ¿ÐµÐ½Ñ Ð² Scilab ÑÐ²Ð»ÑÐµÑÑÑ Ð¾Ð¿ÐµÑÐ°ÑÐ¾ÑÐ¾Ð¼ Ñ Ð°ÑÑÐ¾ÑÐ¸Ð°ÑÐ¸Ð²Ð½Ð¾ÑÑÑÑ + ÑÐ¿ÑÐ°Ð²Ð°, Ð² Ð¾ÑÐ»Ð¸ÑÐ¸Ðµ Ð¾Ñ MatlabÂ® Ð¸ Octave. + ÐÐ°Ð¿ÑÐ¸Ð¼ÐµÑ 2^3^4 Ð² Scilab ÑÐ°Ð²Ð½Ð¾ 2^(3^4), + Ð° Ð² MatlabÂ® Ð¸ Octave ÑÐ°Ð²Ð½Ð¾ (2^3)^4. + + + + + ÐÑÐ¸Ð¼ÐµÑÑ @@ -98,8 +215,27 @@ s^(1:10) exp + expm + + ÐºÑÑÑÐµÑÐºÐ° + + inv + + + ÐÑÑÐ¾ÑÐ¸Ñ + + + 6.0.0 + + Ð¡ Ð´ÐµÑÑÑÐ¸ÑÐ½ÑÐ¼ Ð¸Ð»Ð¸ ÐºÐ¾Ð¼Ð¿Ð»ÐµÐºÑÐ½ÑÐ¼ ÑÐ¸ÑÐ»Ð°Ð¼Ð¸ scalar ^ squareMat + ÑÐµÐ¿ÐµÑÑ Ð´Ð°ÑÑ expm(log(scalar)*squareMat) Ð²Ð¼ÐµÑÑÐ¾ + scalar .^ squareMat. + + + + -- 1.7.9.5