# sparse_vector_triangular_solve_dense_double(_:_:_:_:_:) Solves the system of equations *x = alpha * T⁻¹ * x* for x where *x* is a dense vector and *T* is a triangular sparse matrix, with all operands containing double-precision values. ``` func sparse_vector_triangular_solve_dense_double(_ transt: CBLAS_TRANSPOSE, _ alpha: Double, _ T: sparse_matrix_double!, _ x: UnsafeMutablePointer!, _ incx: sparse_stride) -> sparse_status ``` ## Parameters `transt` Specifies whether to perform the operation with T or the transpose of T. Must be one of `CblasNoTrans` or `CblasTrans`. `alpha` Scalar multiplier of *T*. `T` The sparse triangular matrix. Must be upper or lower triangular matrix. Will return [`SPARSE_ILLEGAL_PARAMETER`](/documentation/Accelerate/SPARSE_ILLEGAL_PARAMETER) if not a triangular matrix. `x` Pointer to the dense vector *x*. The dimension must match the dimension of the triangular matrix *T*. Behavior undefined if this is not met. Negative strides are supported. Note, unlike dense BLAS routines, the pointer points to the last element when stride is negative. On exit holds the solution to the system of equations. `incx` Increment between valid values in the dense vector *x*. Negative strides are supported. ## Return Value On success, [`SPARSE_SUCCESS`](/documentation/Accelerate/SPARSE_SUCCESS) (`x` will be updated with result of the operation). Will return [`SPARSE_ILLEGAL_PARAMETER`](/documentation/Accelerate/SPARSE_ILLEGAL_PARAMETER) if `transt` is invalid or if the matrix `T` is not triangular and `x` will be unchanged. ## Discussion > Important: > Apple provides the BLAS and LAPACK libraries under the Accelerate framework to be in line with LAPACK 3.9.1. Starting with iOS 26, iPadOS 26, macOS 26, tvOS 26, visionOS 26, and watchOS 26, the libraries are in line with LAPACK 3.12.0. These new interfaces provide additional functionality, as well as a new ILP64 interface. To use the new interfaces, define `ACCELERATE_NEW_LAPACK` before including the Accelerate or vecLib headers. For ILP64 interfaces, also define `ACCELERATE_LAPACK_ILP64`. For Swift projects, specify `ACCELERATE_NEW_LAPACK=1` and `ACCELERATE_LAPACK_ILP64=1` as preprocessor macros in Xcode build settings under Apple Clang - Preprocessing > Preprocessor Macros. --- Copyright © 2026 Apple Inc. All rights reserved. | [Terms of Use](https://www.apple.com/legal/internet-services/terms/site.html) | [Privacy Policy](https://www.apple.com/privacy/privacy-policy)