[doc] Update C tutorial. [skip ci] (#8436)

- Use rst references instead of doxygen links.
- Replace deprecated functions.

- Add SaveModel; put free step last [skip ci]

Co-authored-by: Hyunsu Cho <chohyu01@cs.washington.edu>

doc/tutorials/c_api_tutorial.rst

@@ -163,7 +163,7 @@ c. Assertion technique: It works both in C/ C++. If expression evaluates to 0 (f
 Sample examples along with Code snippet to use C API functions
 **************************************************************
 
-1. If the dataset is available in a file, it can be loaded into a ``DMatrix`` object using the `XGDMatrixCreateFromFile <https://xgboost.readthedocs.io/en/stable/dev/c__api_8h.html#a357c3654a1a4dcc05e6b5c50acd17105>`_
+1. If the dataset is available in a file, it can be loaded into a ``DMatrix`` object using the :cpp:func:`XGDMatrixCreateFromFile`
 
 .. code-block:: c
 
@@ -172,7 +172,7 @@ Sample examples along with Code snippet to use C API functions
   safe_xgboost(XGDMatrixCreateFromFile("/path/to/file/filename", silent, &data));
 
 
-2. You can also create a ``DMatrix`` object from a 2D Matrix using the `XGDMatrixCreateFromMat function <https://xgboost.readthedocs.io/en/stable/dev/c__api_8h.html#a079f830cb972df70c7f50fb91678d62f>`_
+2. You can also create a ``DMatrix`` object from a 2D Matrix using the :cpp:func:`XGDMatrixCreateFromMat`
 
 .. code-block:: c
 
@@ -191,7 +191,7 @@ Sample examples along with Code snippet to use C API functions
   safe_xgboost(XGDMatrixCreateFromMat(data2, ROWS, COLS, -1, &dmatrix2));
 
 
-3. Create a Booster object for training & testing on dataset using `XGBoosterCreate <https://xgboost.readthedocs.io/en/stable/dev/c__api_8h.html#ad9fe6f8c8c4901db1c7581a96a21f9ae>`_
+3. Create a Booster object for training & testing on dataset using :cpp:func:`XGBoosterCreate`
 
 .. code-block:: c
 
@@ -202,7 +202,7 @@ Sample examples along with Code snippet to use C API functions
   safe_xgboost(XGBoosterCreate(eval_dmats, eval_dmats_size, &booster));
 
 
-4. For each ``DMatrix`` object, set the labels using `XGDMatrixSetFloatInfo <https://xgboost.readthedocs.io/en/stable/dev/c__api_8h.html#aef75cda93db3ae9af89e465ae7e9cbe3>`_. Later you can access the label using `XGDMatrixGetFloatInfo <https://xgboost.readthedocs.io/en/stable/dev/c__api_8h.html#ab0ee317539a1fb1ce2b5f249e8c768f6>`_.
+4. For each ``DMatrix`` object, set the labels using :cpp:func:`XGDMatrixSetFloatInfo`. Later you can access the label using :cpp:func:`XGDMatrixGetFloatInfo`.
 
 .. code-block:: c
 
@@ -235,7 +235,7 @@ Sample examples along with Code snippet to use C API functions
   }
 
 
-5. Set the parameters for the ``Booster`` object according to the requirement using `XGBoosterSetParam <https://xgboost.readthedocs.io/en/stable/dev/c__api_8h.html#af7378865b0c999d2d08a5b16483b8bcb>`_ . Check out the full list of parameters available `here <https://xgboost.readthedocs.io/en/latest/parameter.html>`_ .
+5. Set the parameters for the ``Booster`` object according to the requirement using :cpp:func:`XGBoosterSetParam` . Check out the full list of parameters available :doc:`here </parameter>` .
 
 .. code-block :: c
 
@@ -247,7 +247,7 @@ Sample examples along with Code snippet to use C API functions
     safe_xgboost(XGBoosterSetParam(booster, "eta", "0.1"));
 
 
-6. Train & evaluate the model using `XGBoosterUpdateOneIter <https://xgboost.readthedocs.io/en/stable/dev/c__api_8h.html#a13594d68b27327db290ec5e0a0ac92ae>`_ and `XGBoosterEvalOneIter <https://xgboost.readthedocs.io/en/stable/dev/c__api_8h.html#a201b53edb9cc52e9def1ccea951d18fe>`_ respectively.
+6. Train & evaluate the model using :cpp:func:`XGBoosterUpdateOneIter` and :cpp:func:`XGBoosterEvalOneIter` respectively.
 
 .. code-block:: c
 
@@ -264,32 +264,31 @@ Sample examples along with Code snippet to use C API functions
       printf("%s\n", eval_result);
     }
 
-.. note:: For customized loss function, use `XGBoosterBoostOneIter function <https://xgboost.readthedocs.io/en/stable/dev/c__api_8h.html#afd4a42c38cfb16d2cf2a9cf5daba4e83>`_ instead and manually specify the gradient and 2nd order gradient.
+.. note:: For customized loss function, use :cpp:func:`XGBoosterBoostOneIter` instead and manually specify the gradient and 2nd order gradient.
 
 
-7.  Predict the result on a test set using `XGBoosterPredict <https://xgboost.readthedocs.io/en/stable/dev/c__api_8h.html#adc14afaedd5f1add105d18942a4de33c>`_
+7.  Predict the result on a test set using :cpp:func:`XGBoosterPredictFromDMatrix`
 
 .. code-block:: c
 
-    bst_ulong output_length;
+    char const config[] =
-
+        "{\"training\": false, \"type\": 0, "
-    const float *output_result;
+        "\"iteration_begin\": 0, \"iteration_end\": 0, \"strict_shape\": false}";
-    safe_xgboost(XGBoosterPredict(booster, test, 0, 0, &output_length, &output_result));
+    /* Shape of output prediction */
+    uint64_t const* out_shape;
+    /* Dimension of output prediction */
+    uint64_t out_dim;
+    /* Pointer to a thread local contigious array, assigned in prediction function. */
+    float const* out_result = NULL;
+    safe_xgboost(
+        XGBoosterPredictFromDMatrix(booster, dmatrix, config, &out_shape, &out_dim, &out_result));
 
     for (unsigned int i = 0; i < output_length; i++){
       printf("prediction[%i] = %f \n", i, output_result[i]);
     }
 
 
-8. Free all the internal structure used in your code using `XGDMatrixFree <https://xgboost.readthedocs.io/en/stable/dev/c__api_8h.html#af06a15433b01e3b8297930a38155e05d>`_ and `XGBoosterFree <https://xgboost.readthedocs.io/en/stable/dev/c__api_8h.html#a5d816936b005a103f0deabf287a6a5da>`_. This step is important to prevent memory leak.
+8. Get the number of features in your dataset using :cpp:func:`XGBoosterGetNumFeature`.
-
-.. code-block:: c
-
-  safe_xgboost(XGDMatrixFree(dmatrix));
-  safe_xgboost(XGBoosterFree(booster));
-
-
-9. Get the number of features in your dataset using `XGBoosterGetNumFeature <https://xgboost.readthedocs.io/en/stable/dev/c__api_8h.html#aa2c22f65cf2770c0e2e56cc7929a14af>`_.
 
 .. code-block:: c
 
@@ -304,12 +303,22 @@ Sample examples along with Code snippet to use C API functions
     printf("num_feature: %lu\n", (unsigned long)(num_of_features));
 
 
-10. Load the model using `XGBoosterLoadModel function <https://xgboost.readthedocs.io/en/stable/dev/c__api_8h.html#a054571e6364f9a1cbf6b6b4fd2f156d6>`_
+
+9. Save the model using :cpp:func:`XGBoosterSaveModel`
 
 .. code-block:: c
 
     BoosterHandle booster;
-    const char *model_path = "/path/of/model";
+    const char *model_path = "/path/of/model.json";
+    safe_xgboost(XGBoosterSaveModel(booster, model_path));
+
+
+10. Load the model using :cpp:func:`XGBoosterLoadModel`
+
+.. code-block:: c
+
+    BoosterHandle booster;
+    const char *model_path = "/path/of/model.json";
 
     // create booster handle first
     safe_xgboost(XGBoosterCreate(NULL, 0, &booster));
@@ -320,3 +329,11 @@ Sample examples along with Code snippet to use C API functions
     safe_xgboost(XGBoosterLoadModel(booster, model_path));
 
     // predict the model here
+
+
+11. Free all the internal structure used in your code using :cpp:func:`XGDMatrixFree` and :cpp:func:`XGBoosterFree`. This step is important to prevent memory leak.
+
+.. code-block:: c
+
+  safe_xgboost(XGDMatrixFree(dmatrix));
+  safe_xgboost(XGBoosterFree(booster));