---
title: "sparse_vector_triangular_solve_dense_float(_:_:_:_:_:)"
framework: accelerate
role: symbol
role_heading: Function
path: "accelerate/sparse_vector_triangular_solve_dense_float(_:_:_:_:_:)"
---

# sparse_vector_triangular_solve_dense_float(_:_:_:_:_:)

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 single-precision values.

## Declaration

```swift
func sparse_vector_triangular_solve_dense_float(_ transt: CBLAS_TRANSPOSE, _ alpha: Float, _ T: sparse_matrix_float!, _ x: UnsafeMutablePointer<Float>!, _ 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 doc://com.apple.accelerate/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

Return Value On success, SPARSE_SUCCESS (x will be updated with result of the  operation).  Will return SPARSE_ILLEGAL_PARAMETER if transt is invalid or if the matrix T is not triangular and x will be unchanged.

## Discussion

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.

## See Also

### Matrix-Vector Operations

- [sparse_matrix_vector_product_dense_double(_:_:_:_:_:_:_:)](accelerate/sparse_matrix_vector_product_dense_double(_:_:_:_:_:_:_:).md)
- [sparse_matrix_vector_product_dense_float(_:_:_:_:_:_:_:)](accelerate/sparse_matrix_vector_product_dense_float(_:_:_:_:_:_:_:).md)
- [sparse_vector_triangular_solve_dense_double(_:_:_:_:_:)](accelerate/sparse_vector_triangular_solve_dense_double(_:_:_:_:_:).md)
- [sparse_outer_product_dense_double(_:_:_:_:_:_:_:_:_:)](accelerate/sparse_outer_product_dense_double(_:_:_:_:_:_:_:_:_:).md)
- [sparse_outer_product_dense_float(_:_:_:_:_:_:_:_:_:)](accelerate/sparse_outer_product_dense_float(_:_:_:_:_:_:_:_:_:).md)
- [sparse_permute_rows_double(_:_:)](accelerate/sparse_permute_rows_double(_:_:).md)
- [sparse_permute_rows_float(_:_:)](accelerate/sparse_permute_rows_float(_:_:).md)
- [sparse_permute_cols_double(_:_:)](accelerate/sparse_permute_cols_double(_:_:).md)
- [sparse_permute_cols_float(_:_:)](accelerate/sparse_permute_cols_float(_:_:).md)
- [sparse_elementwise_norm_double(_:_:)](accelerate/sparse_elementwise_norm_double(_:_:).md)
- [sparse_elementwise_norm_float(_:_:)](accelerate/sparse_elementwise_norm_float(_:_:).md)
- [sparse_operator_norm_double(_:_:)](accelerate/sparse_operator_norm_double(_:_:).md)
- [sparse_operator_norm_float(_:_:)](accelerate/sparse_operator_norm_float(_:_:).md)
- [sparse_matrix_trace_double(_:_:)](accelerate/sparse_matrix_trace_double(_:_:).md)
- [sparse_matrix_trace_float(_:_:)](accelerate/sparse_matrix_trace_float(_:_:).md)