# Matrix derivative

### Description

Evaluate the derivative of the following types (*d* denotes the "curly d" in partial derivatives):

Type 0: scalar-by-scalar

*dy/dx*Type 1: (gradient) scalar-by-vector

*dy/d**x*Type 2: vector-by-scalar

*d**y**/dx*Type 3: (Jacobian) vector-by-vector

*d**y**/d**x*Type 4: scalar-by-matrix

*dy/d**X*Type 5: matrix-by-scalar

*d**Y**/dx*

Note that this is only for real variables although one may modify the source code allowing for some complex-valued ones. Yet correct result is not guaranteed.

### Usage

1 | ```
matrix.derv(Y, X0, order = 1, rel.tol=.Machine$double.eps^0.5)
``` |

### Arguments

`Y` |
One single function (for type 5) whose derivative to be evaluated. YFor a vector or a matrix of functions, the variables and their order must be the same across all functions. Also see Examples. |

`X0` |
Evaluation points of the derivative. It can be a scalar (type 4). XNote that (1) the number of elements in X0 must match the number of variables in each of the functions in Y and, (2) the order of the elements in X0 follows that of the variables in each of the functions in Y. Also see Examples. |

`order` |
The order of the derivative. Must be a non-zero natural number. |

`rel.tol` |
Relative tolerance of the R built-in function |

### Details

Error or misleading result may occur if arugments are not properly supplied. Some examples:

Variables in the functions are not the same: If

`Y=c(function(x,y,z){x^2+y^2*z},`

`function(x,z,y){sqrt(x*y*z)},function(x,y,z){x^y*log(z)})`, the second function as the order of the variables as`x,z,y`.The number of evaluation points and the number of variables not match: If

`Y=matrix(function(x){x^3},function(x){x^2},function(x){x},function(x){1},ncol=2)`and`X0=c(1,2)`, this is a matrix-by-scalar derivative (type 5) but`X0`here is a vector.The order of the variables and the order of the evaluation points: If

`Y=function(a,b,c,d){a*b+c/d}`and`X0=matrix(1:4,ncol=2)`, then the derivative is evaluated at`a=1, b=2, c=3, d=4`respectively.

### Value

`result` |
A matrix even if type 0 (scalar-by-scalar). |

`type` |
The type of derivative: 0,1,...,5 |

### Author(s)

Char Leung

### See Also

`derv`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 | ```
#Type-0 : scalar-by-scalar
X<-3
Y<-function(x){x^3}
matrix.derv(Y,X)
#Type-1 (gradient): scalar-by-vector
X<-c(1,2,0.4)
Y<-function(a,b,c){a^2+b*c}
matrix.derv(Y,X)
#Type-2 : vector-by-scalar
X<-10
Y<-c(function(a){sin(a)},function(a){cos(a)*a^2})
matrix.derv(Y,X)
#Type-3 (Jacobian) : vector-by-vector
X<-c(1,3,5)
Y<-c(function(a,b,c){a^2+c/b},function(a,b,c){a+b/c})
matrix.derv(Y,X)
#Type-4 : scalar-by-matrix
X<-matrix(1:9,ncol=3)
Y<-function(a,b,c,x,y,z,s,t,r){a*b+c-x*y*z*(s+t+r)}
matrix.derv(Y,X)
#Type-5 : matrix-by-scalar
X<-2
Y<-matrix(c(function(a){a^3},function(a){a^2},function(a){a},function(a){1}),ncol=2)
matrix.derv(Y,X)
``` |