update comments and docs

Author: arcadiaphy

src/operator/linalg.h

@@ -228,7 +228,7 @@ void linalg_getri(const Tensor<xpu, 2, DType>& LU,
 
 // Note that this function only implements GPU version with "getriBatched" in cuBLAS.
 // Unlike lapack routines in cpu, it is computed out-of-place, so the final matrix
-// inversion is stored in A.
+// inverse is stored in A.
 template<typename xpu, typename DType>
 void linalg_batch_getri(const Tensor<xpu, 3, DType>& A,
                         const Tensor<xpu, 3, DType>& LU,
@@ -237,7 +237,7 @@ void linalg_batch_getri(const Tensor<xpu, 3, DType>& A,
 
 //////////////////////////////// INVERSE ////////////////////////////////////////////
 
-// CPU/GPU-versions of matrix inversion combining LAPACK function "getrf" and "getri"
+// CPU/GPU-versions of matrix inverse combining LAPACK function "getrf" and "getri"
 // Note that A = inverse(B)
 template<typename xpu, typename DType>
 void linalg_batch_inverse(const Tensor<xpu, 3, DType>& A,

src/operator/linalg_impl.h

@@ -1429,7 +1429,7 @@ LINALG_GPU_BATCH_GETRI(DgetriBatched, double)
 
 //////////////////////////////// INVERSE ////////////////////////////////////////////
 
-// CPU/GPU-versions of matrix inversion combining LAPACK function "getrf" and "getri"
+// CPU/GPU-versions of matrix inverse combining LAPACK function "getrf" and "getri"
 
 // Note A = inverse(B)
 #define LINALG_CPU_BATCH_INVERSE(xpu, DType) \
@@ -1490,7 +1490,7 @@ template<> inline \
 void linalg_batch_inverse<xpu, DType>(const Tensor<xpu, 3, DType>& A, \
                                       const Tensor<xpu, 3, DType>& B, \
                                       const mxnet::OpContext& ctx) { \
-  LOG(FATAL) << "gpu matrix inversion requires CUDA version >= 8.0!"; \
+  LOG(FATAL) << "gpu matrix inverse requires CUDA version >= 8.0!"; \
 }
 
 #endif  // CUDA_VERSION >= 8000
@@ -1552,7 +1552,7 @@ void linalg_batch_det_backward_helper<xpu, DType>(const Tensor<xpu, 3, DType>& L
                                          const Tensor<xpu, 3, DType>& temp, \
                                          const DType zero_det, \
                                          const mxnet::OpContext& ctx) { \
-  LOG(FATAL) << "gpu matrix inversion requires CUDA version >= 8.0!"; \
+  LOG(FATAL) << "gpu matrix inverse requires CUDA version >= 8.0!"; \
 }
 
 #endif  // CUDA_VERSION >= 8000

src/operator/tensor/la_op.cc

@@ -952,18 +952,18 @@ If *n>2*, *det* is performed separately on the trailing two dimensions
 for all inputs (batch mode).
 
 .. note:: The operator supports float32 and float64 data types only.
-.. note:: There is no gradient backwarded when det(A) == 0 because it's
-          rarely hit upon in float point computation and the Jacobi's
-          formula on determinant gradient is not computationally efficient
-          when A is non-invertible.
+.. note:: There is no gradient backwarded when A is non-invertible (which is
+          equivalent to det(A) = 0) because zero is rarely hit upon in float
+          point computation and the Jacobi's formula on determinant gradient
+          is not computationally efficient when A is non-invertible.
 
 Examples::
 
-   Single matrix inversion
+   Single matrix determinant
    A = [[1., 4.], [2., 3.]]
    det(A) = [-5.]
 
-   Batch matrix inversion
+   Batch matrix determinant
    A = [[[1., 4.], [2., 3.]],
         [[2., 3.], [1., 4.]]]
    det(A) = [-5., 5.]
@@ -1006,16 +1006,18 @@ for all inputs (batch mode).
 .. note:: The operator supports float32 and float64 data types only.
 .. note:: The gradient is not properly defined on sign, so the gradient of
           it is not backwarded.
+.. note:: No gradient is backwarded when A is non-invertible. Please see
+          the docs of operator det for detail.
 
 Examples::
 
-   Single matrix inversion
+   Single matrix signed log determinant
    A = [[2., 3.], [1., 4.]]
    sign, logabsdet = slogdet(A)
    sign = [1.]
    logabsdet = [1.609438]
 
-   Batch matrix inversion
+   Batch matrix signed log determinant
    A = [[[2., 3.], [1., 4.]],
         [[1., 2.], [2., 4.]],
         [[1., 2.], [4., 3.]]]