# W3cubDocs

/TensorFlow 2.4

Computes the GRU cell back-propagation for 1 time step.

Args x: Input to the GRU cell. h_prev: State input from the previous GRU cell. w_ru: Weight matrix for the reset and update gate. w_c: Weight matrix for the cell connection gate. b_ru: Bias vector for the reset and update gate. b_c: Bias vector for the cell connection gate. r: Output of the reset gate. u: Output of the update gate. c: Output of the cell connection gate. d_h: Gradients of the h_new wrt to objective function.

Returns d_x: Gradients of the x wrt to objective function. d_h_prev: Gradients of the h wrt to objective function. d_c_bar Gradients of the c_bar wrt to objective function. d_r_bar_u_bar Gradients of the r_bar & u_bar wrt to objective function.

This kernel op implements the following mathematical equations:

Note on notation of the variables:

Concatenation of a and b is represented by a_b Element-wise dot product of a and b is represented by ab Element-wise dot product is represented by \circ Matrix multiplication is represented by *

`w_ru` can be segmented into 4 different matrices.

```w_ru = [w_r_x w_u_x
w_r_h_prev w_u_h_prev]
```

Similarly, `w_c` can be segmented into 2 different matrices.

```w_c = [w_c_x w_c_h_prevr]
```

Same goes for biases.

```b_ru = [b_ru_x b_ru_h]
b_c = [b_c_x b_c_h]
```

Another note on notation:

```d_x = d_x_component_1 + d_x_component_2

where d_x_component_1 = d_r_bar * w_r_x^T + d_u_bar * w_r_x^T
and d_x_component_2 = d_c_bar * w_c_x^T

d_h_prev = d_h_prev_component_1 + d_h_prevr \circ r + d_h \circ u
where d_h_prev_componenet_1 = d_r_bar * w_r_h_prev^T + d_u_bar * w_r_h_prev^T
```

```d_c_bar = d_h \circ (1-u) \circ (1-c \circ c)
d_u_bar = d_h \circ (h-c) \circ u \circ (1-u)

d_r_bar_u_bar = [d_r_bar d_u_bar]

[d_x_component_1 d_h_prev_component_1] = d_r_bar_u_bar * w_ru^T

[d_x_component_2 d_h_prevr] = d_c_bar * w_c^T

d_x = d_x_component_1 + d_x_component_2

d_h_prev = d_h_prev_component_1 + d_h_prevr \circ r + u
```

Below calculation is performed in the python wrapper for the Gradients (not in the gradient kernel.)

```d_w_ru = x_h_prevr^T * d_c_bar

d_w_c = x_h_prev^T * d_r_bar_u_bar

d_b_ru = sum of d_r_bar_u_bar along axis = 0

d_b_c = sum of d_c_bar along axis = 0
```
Args
`x` A `Tensor`. Must be one of the following types: `float32`.
`h_prev` A `Tensor`. Must have the same type as `x`.
`w_ru` A `Tensor`. Must have the same type as `x`.
`w_c` A `Tensor`. Must have the same type as `x`.
`b_ru` A `Tensor`. Must have the same type as `x`.
`b_c` A `Tensor`. Must have the same type as `x`.
`r` A `Tensor`. Must have the same type as `x`.
`u` A `Tensor`. Must have the same type as `x`.
`c` A `Tensor`. Must have the same type as `x`.
`d_h` A `Tensor`. Must have the same type as `x`.
`name` A name for the operation (optional).
Returns
A tuple of `Tensor` objects (d_x, d_h_prev, d_c_bar, d_r_bar_u_bar).
`d_x` A `Tensor`. Has the same type as `x`.
`d_h_prev` A `Tensor`. Has the same type as `x`.
`d_c_bar` A `Tensor`. Has the same type as `x`.
`d_r_bar_u_bar` A `Tensor`. Has the same type as `x`.