taylor expansion, well defined

can directly write ODE solution:

dx/dt = Ax

x = exp(tA)x_0

if A is diagonalizable, A = PvP^-1, v is diagonal matrix

exp(A) = P exp(v) P^-1

inverse is exp(-A)

A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is 0.

transformations don’t affect topology, are called homeomorphisms

tanh (and sigmoid and softplus but not ReLU) are continuous functions with continuous inverses

Qs:

how stiff

how many vars

how accrate

how sensitive(chaos)

how well-behaved is f(x,t)

how costly is f(x,t) and its jacobian

implicit,

more stable, solving stiff problems requires implicit methods

Euler’s method

trapezoidal

Newton-Raphson

stability criterion

x’(t) = a x(t)

forward euler is conditionally stable

backward euler is unconditionally stable

truncation error

Runge-Kutta methods

stage order

Multi-step methods

predictor-corrector method

richardson extrapolation

automatic selection

[Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations | SIAM Journal on Scientific and Statistical Computing | Vol. 4, No. 1 | Society for Industrial and Applied Mathematics](https://epubs.siam.org/doi/pdf/10.1137/0904010?casa_token=sBjDZTSayFQAAAAA:XhlfyWkS4MRFNRnrZ6LmQff_UXAH7riLBkpcA58llDnYEJycmMMbMCli9cFkoYKRT7uNos94IpA) |

instability

zero stability

Re(lambda) < 0, x’(t) = lambda x(t) numerical solution decays to 0

region of absolute stability z = \lambda \delta t

all scaled eigenvalues of Jacobian should be in region, for ODE system

stability region contains the entire lefthalf plane

backward euler, implicit midpoint

no explicit one-step method can be A-stable

All explicit RK methods with r stages and of order r have the same stability region

if it is A-stable and it damps fast componentsof the solution

A stiff problem is one where ∆t has to be small even though the solution is smooth and a large ∆t is OK for accuracy

if the solution evolves on widely-separated timescales and the fast time scale decays (dies out) quickly

given linear ODE system, x’(t) = A x(t), decompose A to separate parts, x is formed by uncoupled n different y variables, each of n ODEs(y) is independent of the others

timestep of original system must be smaller than the smallest stability limits

the system is stiff if a strong separation of stability time scale (eigenvalue ratio)

Jacobian for non-linear system, complex eigenvalues

adjoint method

auto diff

半格是满足运算是幂等的和交换的半群。

半群是闭合于结合性二元运算之下的集合 S 构成的代数结构

集合S和其上的二元运算·:S×S→S。若·满足结合律，即：∀x,y,z∈S，有(x·y)·z=x·(y·z)，则称有序对(S,·)为半群

**history monoid** is a way of representing the histories of concurrently running computer processes as a collection of strings, each string representing the individual history of a process

考虑任意一个偏序集合（L,≤），如果对集合L中的任意元素a,b，使得a,b在L中存在一个最大下界，和最小上界，则(L,≤)是一个格

一个格是完全的，如果它的所有子集都有一个交和一个并

a collection of random variables is heteroscedastic (or heteroskedastic;[a] from Ancient Greek hetero “different” and skedasis “dispersion”) if there are sub-populations that have different variabilities from others