diff --git a/nalgebra-sparse/src/cs.rs b/nalgebra-sparse/src/cs.rs
index 30bdc9b8..809fb276 100644
--- a/nalgebra-sparse/src/cs.rs
+++ b/nalgebra-sparse/src/cs.rs
@@ -4,7 +4,7 @@ use num_traits::One;
 
 use nalgebra::Scalar;
 
-use crate::pattern::SparsityPattern;
+use crate::pattern::{BuilderInsertError, SparsityPattern, SparsityPatternBuilder};
 use crate::utils::{apply_permutation, compute_sort_permutation};
 use crate::{SparseEntry, SparseEntryMut, SparseFormatError, SparseFormatErrorKind};
 
@@ -700,3 +700,56 @@ where
 
     Ok(())
 }
+
+#[derive(Debug, Clone, PartialEq, Eq)]
+pub struct CsBuilder<T> {
+    sparsity_builder: SparsityPatternBuilder,
+    values: Vec<T>,
+}
+
+impl<T> CsBuilder<T> {
+    /// Constructs a new CsBuilder of the given size.
+    pub fn new(major_dim: usize, minor_dim: usize) -> Self {
+        Self {
+            sparsity_builder: SparsityPatternBuilder::new(major_dim, minor_dim),
+            values: vec![],
+        }
+    }
+    /// Given an existing CsMatrix, allows for modification by converting it into a builder.
+    pub fn from_mat(mat: CsMatrix<T>) -> Self {
+        let CsMatrix {
+            sparsity_pattern,
+            values,
+        } = mat;
+
+        CsBuilder {
+            sparsity_builder: SparsityPatternBuilder::from(sparsity_pattern),
+            values,
+        }
+    }
+    /// Backtracks to a given major index
+    pub fn revert_to_major(&mut self, maj: usize) -> bool {
+        if !self.sparsity_builder.revert_to_major(maj) {
+            return false;
+        }
+
+        self.values.truncate(self.sparsity_builder.num_entries());
+        true
+    }
+    pub fn insert(&mut self, maj: usize, min: usize, val: T) -> Result<(), BuilderInsertError> {
+        self.sparsity_builder.insert(maj, min)?;
+        self.values.push(val);
+        Ok(())
+    }
+    pub fn build(self) -> CsMatrix<T> {
+        let CsBuilder {
+            sparsity_builder,
+            values,
+        } = self;
+        let sparsity_pattern = sparsity_builder.build();
+        CsMatrix {
+            sparsity_pattern,
+            values,
+        }
+    }
+}
diff --git a/nalgebra-sparse/src/csc.rs b/nalgebra-sparse/src/csc.rs
index 8d6615a9..201e5633 100644
--- a/nalgebra-sparse/src/csc.rs
+++ b/nalgebra-sparse/src/csc.rs
@@ -7,12 +7,14 @@
 mod csc_serde;
 
 use crate::cs;
-use crate::cs::{CsLane, CsLaneIter, CsLaneIterMut, CsLaneMut, CsMatrix};
+use crate::cs::{CsBuilder, CsLane, CsLaneIter, CsLaneIterMut, CsLaneMut, CsMatrix};
 use crate::csr::CsrMatrix;
-use crate::pattern::{SparsityPattern, SparsityPatternFormatError, SparsityPatternIter};
+use crate::pattern::{
+    BuilderInsertError, SparsityPattern, SparsityPatternFormatError, SparsityPatternIter,
+};
 use crate::{SparseEntry, SparseEntryMut, SparseFormatError, SparseFormatErrorKind};
 
-use nalgebra::Scalar;
+use nalgebra::{RealField, Scalar};
 use num_traits::One;
 use std::slice::{Iter, IterMut};
 
@@ -553,6 +555,269 @@ impl<T> CscMatrix<T> {
         self.filter(|i, j, _| i >= j)
     }
 
+    /// Solves a lower triangular system, `self` is a matrix of NxN, and `b` is a column vector of size N
+    /// Assuming that b is dense.
+    pub fn dense_lower_triangular_solve(&self, b: &[T], out: &mut [T], unit_diagonal: bool)
+    where
+        T: RealField + Copy,
+    {
+        assert_eq!(self.nrows(), self.ncols());
+        assert_eq!(self.ncols(), b.len());
+        assert_eq!(out.len(), b.len());
+        out.copy_from_slice(b);
+        let n = b.len();
+
+        for i in 0..n {
+            let col = self.col(i);
+            let mut iter = col.row_indices().iter().zip(col.values().iter()).peekable();
+            while iter.next_if(|n| *n.0 < i).is_some() {}
+            if let Some(n) = iter.peek() {
+                if *n.0 == i && !unit_diagonal {
+                    assert!(*n.0 <= i);
+                    out[i] /= *n.1;
+                    iter.next();
+                }
+            }
+            let mul = out[i];
+            for (&ri, &v) in col.row_indices().iter().zip(col.values().iter()) {
+                use std::cmp::Ordering::*;
+                // ensure that only using the lower part
+                match ri.cmp(&i) {
+                    Greater => out[ri] -= v * mul,
+                    Equal | Less => {}
+                }
+            }
+        }
+    }
+
+    /// Solves an upper triangular system, `self` is a matrix of NxN, and `b` is a column vector of size N
+    /// Assuming that b is dense.
+    pub fn dense_upper_triangular_solve(&self, b: &[T], out: &mut [T])
+    where
+        T: RealField + Copy,
+    {
+        assert_eq!(self.nrows(), self.ncols());
+        assert_eq!(self.ncols(), b.len());
+        assert_eq!(out.len(), b.len());
+        out.copy_from_slice(b);
+        let n = b.len();
+
+        for i in (0..n).rev() {
+            let col = self.col(i);
+            let mut iter = col
+                .row_indices()
+                .iter()
+                .zip(col.values().iter())
+                .rev()
+                .peekable();
+            while iter.next_if(|n| *n.0 > i).is_some() {}
+            if let Some(n) = iter.peek() {
+                if *n.0 == i {
+                    out[i] /= *n.1;
+                    iter.next();
+                }
+            }
+            // introduce a NaN, intentionally, if the diagonal doesn't have a value.
+            let mul = out[i];
+            for (&row, &v) in iter {
+                use std::cmp::Ordering::*;
+                match row.cmp(&i) {
+                    Less => out[row] -= v * mul,
+                    Equal | Greater => {}
+                }
+            }
+        }
+    }
+
+    /// Solves a sparse lower triangular system `Ax = b`, with both the matrix and vector
+    /// sparse.
+    /// sparsity_idxs should be precomputed using the sparse_lower_triangle.
+    pub fn sparse_upper_triangular_solve_sorted(
+        &self,
+        b_idxs: &[usize],
+        b: &[T],
+
+        out_sparsity_pattern: &[usize],
+        out: &mut [T],
+    ) where
+        T: RealField + Copy,
+    {
+        assert_eq!(self.nrows(), self.ncols());
+        assert_eq!(b_idxs.len(), b.len());
+        assert!(b_idxs.iter().all(|&bi| bi < self.ncols()));
+
+        assert_eq!(out_sparsity_pattern.len(), out.len());
+        assert!(out_sparsity_pattern.iter().all(|&bi| bi < self.ncols()));
+
+        // initialize out with b
+        out.fill(T::zero());
+        for (&bv, &bi) in b.iter().zip(b_idxs.iter()) {
+            let out_pos = out_sparsity_pattern.iter().position(|&p| p == bi).unwrap();
+            out[out_pos] = bv;
+        }
+
+        for (i, &row) in out_sparsity_pattern.iter().enumerate().rev() {
+            let col = self.col(row);
+            let mut iter = col
+                .row_indices()
+                .iter()
+                .zip(col.values().iter())
+                .rev()
+                .peekable();
+
+            while iter.next_if(|n| *n.0 > row).is_some() {}
+            match iter.peek() {
+                Some((&r, &l_val)) if r == row => out[i] /= l_val,
+                // here it now becomes implicitly 0,
+                // likely this should introduce NaN or some other behavior.
+                _ => {}
+            }
+            let mul = out[i];
+            for (ni, &nrow) in out_sparsity_pattern[i + 1..].iter().enumerate().rev() {
+                assert!(nrow > row);
+                while iter.next_if(|n| *n.0 > nrow).is_some() {}
+                let l_val = match iter.peek() {
+                    Some((&r, &l_val)) if r == nrow => l_val,
+                    _ => continue,
+                };
+                out[ni] -= l_val * mul;
+            }
+        }
+    }
+
+    /// Solves a sparse lower triangular system `Ax = b`, with both the matrix and vector
+    /// sparse.
+    /// sparsity_idxs should be precomputed using the sparse_lower_triangle.
+    /// Assumes that the diagonal of the sparse matrix is all 1 if `assume_unit` is true.
+    pub fn sparse_lower_triangular_solve(
+        &self,
+        b_idxs: &[usize],
+        b: &[T],
+        // idx -> row
+        // for now, is permitted to be unsorted
+        // TODO maybe would be better to enforce sorted, but would have to sort internally.
+        out_sparsity_pattern: &[usize],
+        out: &mut [T],
+        assume_unit: bool,
+    ) where
+        T: RealField + Copy,
+    {
+        assert_eq!(self.nrows(), self.ncols());
+        assert_eq!(b.len(), b_idxs.len());
+        assert!(b_idxs.iter().all(|&bi| bi < self.ncols()));
+
+        assert_eq!(out_sparsity_pattern.len(), out.len());
+        assert!(out_sparsity_pattern.iter().all(|&i| i < self.ncols()));
+
+        let is_sorted = (0..out_sparsity_pattern.len() - 1)
+            .all(|i| out_sparsity_pattern[i] < out_sparsity_pattern[i + 1]);
+        if is_sorted {
+            return self.sparse_lower_triangular_solve_sorted(
+                b_idxs,
+                b,
+                out_sparsity_pattern,
+                out,
+                assume_unit,
+            );
+        }
+
+        // initialize out with b
+        out.fill(T::zero());
+        for (&bv, &bi) in b.iter().zip(b_idxs.iter()) {
+            let out_pos = out_sparsity_pattern.iter().position(|&p| p == bi).unwrap();
+            out[out_pos] = bv;
+        }
+
+        for (i, &row) in out_sparsity_pattern.iter().enumerate() {
+            let col = self.col(row);
+            if !assume_unit {
+                if let Some(l_val) = col.get_entry(row) {
+                    out[i] /= l_val.into_value();
+                } else {
+                    // diagonal is 0, non-invertible
+                    out[i] /= T::zero();
+                }
+            }
+            let mul = out[i];
+            for (ni, &nrow) in out_sparsity_pattern.iter().enumerate() {
+                if nrow <= row {
+                    continue;
+                }
+                // TODO in a sorted version may be able to iterate without
+                // having the cost of binary search at each iteration
+                let l_val = if let Some(l_val) = col.get_entry(nrow) {
+                    l_val.into_value()
+                } else {
+                    continue;
+                };
+                out[ni] -= l_val * mul;
+            }
+        }
+    }
+    /// Solves a sparse lower triangular system `Ax = b`, with both the matrix and vector
+    /// sparse.
+    /// sparsity_idxs should be precomputed using the sparse_lower_triangle pattern.
+    ///
+    /// `out_sparsity_pattern` must also be pre-sorted.
+    ///
+    /// Assumes that the diagonal of the sparse matrix is all 1 if `assume_unit` is true.
+    pub fn sparse_lower_triangular_solve_sorted(
+        &self,
+        // input vector idxs & values
+        b_idxs: &[usize],
+        b: &[T],
+        // idx -> row
+        // for now, is permitted to be unsorted
+        // TODO maybe would be better to enforce sorted, but would have to sort internally.
+        out_sparsity_pattern: &[usize],
+        out: &mut [T],
+        assume_unit: bool,
+    ) where
+        T: RealField + Copy,
+    {
+        assert_eq!(self.nrows(), self.ncols());
+        assert_eq!(b.len(), b_idxs.len());
+        assert!(b_idxs.iter().all(|&bi| bi < self.ncols()));
+
+        assert_eq!(out_sparsity_pattern.len(), out.len());
+        assert!(out_sparsity_pattern.iter().all(|&i| i < self.ncols()));
+
+        // initialize out with b
+        // TODO can make this more efficient by keeping two iterators in sorted order
+        out.fill(T::zero());
+        for (&bv, &bi) in b.iter().zip(b_idxs.iter()) {
+            let out_pos = out_sparsity_pattern.iter().position(|&p| p == bi).unwrap();
+            out[out_pos] = bv;
+        }
+        // end init
+
+        // assuming that the output sparsity pattern is sorted
+        // iterate thru
+        for (i, &row) in out_sparsity_pattern.iter().enumerate() {
+            let col = self.col(row);
+            let mut iter = col.row_indices().iter().zip(col.values().iter()).peekable();
+            if !assume_unit {
+                while iter.next_if(|n| *n.0 < row).is_some() {}
+                match iter.peek() {
+                    Some((&r, &l_val)) if r == row => out[i] /= l_val,
+                    // here it now becomes implicitly 0,
+                    // likely this should introduce NaN or some other behavior.
+                    _ => {}
+                }
+            }
+            let mul = out[i];
+            for (ni, &nrow) in out_sparsity_pattern.iter().enumerate().skip(i + 1) {
+                assert!(nrow > row);
+                while iter.next_if(|n| *n.0 < nrow).is_some() {}
+                let l_val = match iter.peek() {
+                    Some((&r, &l_val)) if r == nrow => l_val,
+                    _ => continue,
+                };
+                out[ni] -= l_val * mul;
+            }
+        }
+    }
+
     /// Returns the diagonal of the matrix as a sparse matrix.
     #[must_use]
     pub fn diagonal_as_csc(&self) -> Self
@@ -784,3 +1049,30 @@ where
         self.lane_iter.next().map(|lane| CscColMut { lane })
     }
 }
+
+/// An incremental builder for a Csc matrix.
+#[derive(Debug, Clone, PartialEq, Eq)]
+pub struct CscBuilder<T>(CsBuilder<T>);
+
+impl<T> CscBuilder<T> {
+    /// Constructs a new instance of a Csc Builder.
+    pub fn new(rows: usize, cols: usize) -> Self {
+        Self(CsBuilder::new(cols, rows))
+    }
+    /// Convert back from a matrix to a CscBuilder.
+    pub fn from_mat(mat: CscMatrix<T>) -> Self {
+        Self(CsBuilder::from_mat(mat.cs))
+    }
+    /// Backtracks back to column `col`, deleting all entries ahead of it.
+    pub fn revert_to_col(&mut self, col: usize) -> bool {
+        self.0.revert_to_major(col)
+    }
+    /// Inserts a value into the builder. Must be called in ascending col, row order.
+    pub fn insert(&mut self, row: usize, col: usize, val: T) -> Result<(), BuilderInsertError> {
+        self.0.insert(col, row, val)
+    }
+    /// Converts this builder into a valid CscMatrix.
+    pub fn build(self) -> CscMatrix<T> {
+        CscMatrix { cs: self.0.build() }
+    }
+}
diff --git a/nalgebra-sparse/src/csr.rs b/nalgebra-sparse/src/csr.rs
index d31c86b0..0fcf84c7 100644
--- a/nalgebra-sparse/src/csr.rs
+++ b/nalgebra-sparse/src/csr.rs
@@ -12,7 +12,7 @@ use crate::csc::CscMatrix;
 use crate::pattern::{SparsityPattern, SparsityPatternFormatError, SparsityPatternIter};
 use crate::{SparseEntry, SparseEntryMut, SparseFormatError, SparseFormatErrorKind};
 
-use nalgebra::Scalar;
+use nalgebra::{DMatrix, DMatrixView, RealField, Scalar};
 use num_traits::One;
 
 use std::slice::{Iter, IterMut};
@@ -573,6 +573,55 @@ impl<T> CsrMatrix<T> {
     {
         CscMatrix::from(self).transpose_as_csr()
     }
+
+    /// Solves the equation `Ax = b`, treating `self` as an upper triangular matrix.
+    /// If `A` is not upper triangular, elements in the lower triangle below the diagonal
+    /// will be ignored.
+    ///
+    /// If `m` has zeros along the diagonal, returns `None`.
+    /// Panics:
+    /// Panics if `A` and `b` have incompatible shapes, specifically if `b`
+    /// has a different number of rows and than `a`.
+    pub fn solve_upper_triangular<'a>(&self, b: impl Into<DMatrixView<'a, T>>) -> Option<DMatrix<T>>
+    where
+        T: RealField + Scalar,
+    {
+        // https://www.nicolasboumal.net/papers/MAT321_Lecture_notes_Boumal_2019.pdf
+        // page 48
+        let b: DMatrixView<'a, T> = b.into();
+        assert_eq!(b.nrows(), self.nrows());
+
+        let out_cols = b.ncols();
+        let out_rows = self.nrows();
+
+        let mut out = DMatrix::zeros(out_rows, out_cols);
+        for r in (0..out_rows).rev() {
+            let row = self.row(r);
+            // only take upper triangle elements
+            let mut row_iter = row
+                .col_indices()
+                .iter()
+                .copied()
+                .zip(row.values().iter())
+                .filter(|&(c, _)| c >= r);
+
+            let (c, div) = row_iter.next()?;
+            // This implies there is a 0 on the diagonal
+            if c != r || div.is_zero() {
+                return None;
+            }
+            for c in 0..out_cols {
+                let numer = b.index((r, c)).clone();
+                let numer = numer
+                    - row_iter
+                        .clone()
+                        .map(|(a_col, val)| val.clone() * b.index((a_col, c)).clone())
+                        .fold(T::zero(), |acc, n| acc + n);
+                *out.index_mut((r, c)) = numer / div.clone();
+            }
+        }
+        Some(out)
+    }
 }
 
 impl<T> Default for CsrMatrix<T> {
diff --git a/nalgebra-sparse/src/factorization/lu.rs b/nalgebra-sparse/src/factorization/lu.rs
new file mode 100644
index 00000000..8ca6e6c2
--- /dev/null
+++ b/nalgebra-sparse/src/factorization/lu.rs
@@ -0,0 +1,103 @@
+use crate::SparseEntryMut;
+use crate::{CscBuilder, CscMatrix};
+use nalgebra::{DMatrix, RealField};
+
+/// Constructs an LU Factorization using a left-looking approach.
+/// This means it will construct each column, starting from the leftmost one.
+pub struct LeftLookingLUFactorization<T> {
+    /// A single matrix stores both the lower and upper triangular components
+    l_u: CscMatrix<T>,
+}
+
+impl<T: RealField + Copy> LeftLookingLUFactorization<T> {
+    /// Returns the upper triangular part of this matrix.
+    pub fn u(&self) -> CscMatrix<T> {
+        self.l_u.upper_triangle()
+    }
+
+    /// Returns the joint L\U matrix. Here, `L` implicitly has 1 along the diagonal.
+    pub fn lu(&self) -> &CscMatrix<T> {
+        &self.l_u
+    }
+
+    /// Returns the lower triangular part of this matrix.
+    pub fn l(&self) -> CscMatrix<T> {
+        let mut l = self.l_u.lower_triangle();
+        let n = self.l_u.nrows();
+        for i in 0..n {
+            if let SparseEntryMut::NonZero(v) = l.index_entry_mut(i, i) {
+                *v = T::one();
+            } else {
+                unreachable!();
+            }
+        }
+        l
+    }
+
+    /// Computes `x` in `LUx = b`, where `b` is a dense vector.
+    pub fn solve(&self, b: &[T]) -> DMatrix<T> {
+        let mut y = vec![T::zero(); b.len()];
+        // Implementation: Solve two systems: Ly = b, then Ux = y.
+        self.l_u.dense_lower_triangular_solve(b, &mut y, true);
+        let mut out = y.clone();
+        self.l_u.dense_upper_triangular_solve(&y, &mut out);
+
+        DMatrix::from_vec(b.len(), 1, out)
+    }
+
+    /// Construct a new sparse LU factorization
+    /// from a given CSC matrix.
+    pub fn new(a: &CscMatrix<T>) -> Self {
+        assert_eq!(a.nrows(), a.ncols());
+        let n = a.nrows();
+
+        // this initially starts as an identity  matrix.
+        // but the ones are all implicit.
+        let mut csc_builder = CscBuilder::new(n, n);
+
+        let mut val_buf = vec![];
+        let mut pat_buf = vec![];
+
+        for (ci, col) in a.col_iter().enumerate() {
+            let curr_mat = csc_builder.build();
+
+            curr_mat
+                .pattern()
+                .sparse_lower_triangular_solve(col.row_indices(), &mut pat_buf);
+            pat_buf.sort_unstable();
+            val_buf.resize_with(pat_buf.len(), T::zero);
+
+            // Solve the current column, assuming that it is lower triangular
+            curr_mat.sparse_lower_triangular_solve_sorted(
+                col.row_indices(),
+                col.values(),
+                &pat_buf,
+                &mut val_buf,
+                true,
+            );
+
+            // convert builder back to matrix
+            csc_builder = CscBuilder::from_mat(curr_mat);
+            assert!(csc_builder.revert_to_col(ci));
+            let mut ukk = T::zero();
+            for (row, val) in pat_buf.drain(..).zip(val_buf.drain(..)) {
+                use std::cmp::Ordering;
+                let val = match row.cmp(&ci) {
+                    Ordering::Less => val,
+                    Ordering::Equal => {
+                        ukk = val;
+                        val
+                    }
+                    Ordering::Greater => {
+                        assert_ne!(ukk, T::zero());
+                        val / ukk
+                    }
+                };
+                assert_eq!(csc_builder.insert(row, ci, val), Ok(()));
+            }
+        }
+
+        let l_u = csc_builder.build();
+        Self { l_u }
+    }
+}
diff --git a/nalgebra-sparse/src/factorization/mod.rs b/nalgebra-sparse/src/factorization/mod.rs
index b77a857c..1f474672 100644
--- a/nalgebra-sparse/src/factorization/mod.rs
+++ b/nalgebra-sparse/src/factorization/mod.rs
@@ -2,5 +2,7 @@
 //!
 //! Currently, the only factorization provided here is the [`CscCholesky`] factorization.
 mod cholesky;
+mod lu;
 
 pub use cholesky::*;
+pub use lu::*;
diff --git a/nalgebra-sparse/src/lib.rs b/nalgebra-sparse/src/lib.rs
index a022dedb..53cce200 100644
--- a/nalgebra-sparse/src/lib.rs
+++ b/nalgebra-sparse/src/lib.rs
@@ -171,7 +171,7 @@ use std::error::Error;
 use std::fmt;
 
 pub use self::coo::CooMatrix;
-pub use self::csc::CscMatrix;
+pub use self::csc::{CscBuilder, CscMatrix};
 pub use self::csr::CsrMatrix;
 
 /// Errors produced by functions that expect well-formed sparse format data.
@@ -279,4 +279,11 @@ impl<'a, T: Clone + Zero> SparseEntryMut<'a, T> {
             SparseEntryMut::Zero => T::zero(),
         }
     }
+    /// If the entry is nonzero, returns `Some(&mut value)`, otherwise returns `None`.
+    pub fn nonzero(self) -> Option<&'a mut T> {
+        match self {
+            SparseEntryMut::NonZero(v) => Some(v),
+            SparseEntryMut::Zero => None,
+        }
+    }
 }
diff --git a/nalgebra-sparse/src/pattern.rs b/nalgebra-sparse/src/pattern.rs
index 803e7f2d..36bc4baa 100644
--- a/nalgebra-sparse/src/pattern.rs
+++ b/nalgebra-sparse/src/pattern.rs
@@ -61,6 +61,14 @@ impl SparsityPattern {
             minor_dim,
         }
     }
+    /// Creates the sparsity pattern of an identity matrix of size `n`.
+    pub fn identity(n: usize) -> Self {
+        Self {
+            major_offsets: (0..=n).collect(),
+            minor_indices: (0..n).collect(),
+            minor_dim: n,
+        }
+    }
 
     /// The offsets for the major dimension.
     #[inline]
@@ -109,6 +117,13 @@ impl SparsityPattern {
     pub fn lane(&self, major_index: usize) -> &[usize] {
         self.get_lane(major_index).unwrap()
     }
+    /// Returns an iterator over all lanes in this sparsity pattern, in order.
+    /// Does not omit empty lanes.
+    #[inline]
+    #[must_use]
+    pub fn lanes(&self) -> impl Iterator<Item = &[usize]> {
+        (0..self.major_offsets.len() - 1).map(move |r| self.lane(r))
+    }
 
     /// Get the lane at the given index, or `None` if out of bounds.
     #[inline]
@@ -297,6 +312,198 @@ impl SparsityPattern {
         )
         .expect("Internal error: Transpose should never fail.")
     }
+
+    /// Computes the output sparsity pattern of `x` in `Ax = b`.
+    /// where A's nonzero pattern is given by `self` and the non-zero indices
+    /// of vector `b` are specified as a slice.
+    /// The output is not necessarily in sorted order, but is topological sort order.
+    /// Treats `self` as lower triangular, even if there are elements in the upper triangle.
+    /// Acts as if b is one major lane (i.e. CSC matrix and one column)
+    pub fn sparse_lower_triangular_solve(&self, b: &[usize], out: &mut Vec<usize>) {
+        assert!(b.iter().all(|&i| i < self.major_dim()));
+        out.clear();
+
+        // From a given starting column, traverses and finds all reachable indices.
+        fn reach(sp: &SparsityPattern, j: usize, out: &mut Vec<usize>) {
+            // already traversed
+            if out.contains(&j) {
+                return;
+            }
+
+            out.push(j);
+            for &i in sp.lane(j) {
+                if i < j {
+                    continue;
+                }
+                reach(sp, i, out);
+            }
+        }
+
+        for &i in b {
+            reach(&self, i, out);
+        }
+    }
+
+    /// Computes the output sparsity pattern of `x` in `Ax = b`.
+    /// where A's nonzero pattern is given by `self` and the non-zero indices
+    /// of vector `b` are specified as a slice.
+    /// The output is not necessarily in sorted order, but is topological sort order.
+    /// Treats `self` as upper triangular, even if there are elements in the lower triangle.
+    /// Acts as if b is one major lane (i.e. CSC matrix and one column)
+    pub fn sparse_upper_triangular_solve(&self, b: &[usize], out: &mut Vec<usize>) {
+        assert!(b.iter().all(|&i| i < self.major_dim()));
+        out.clear();
+
+        // From a given starting column, traverses and finds all reachable indices.
+        fn reach(sp: &SparsityPattern, j: usize, out: &mut Vec<usize>) {
+            // already traversed
+            if out.contains(&j) {
+                return;
+            }
+
+            out.push(j);
+            // iteration order here does not matter, but technically it should be rev?
+            for &i in sp.lane(j).iter().rev() {
+                if i > j {
+                    continue;
+                }
+                reach(sp, i, out);
+            }
+        }
+
+        for &i in b {
+            reach(&self, i, out);
+        }
+    }
+
+    /// Left-looking Sparse LU decomposition from Gilbert/Peierls.
+    /// returns the sparsity pattern of the output
+    pub fn left_looking_lu_decomposition(&self) -> SparsityPattern {
+        assert_eq!(self.major_dim(), self.minor_dim());
+        let n = self.minor_dim();
+        let mut sp = SparsityPatternBuilder::new(n, n);
+        let mut x = vec![];
+        for col in 0..n {
+            sp.valid_partial()
+                .sparse_lower_triangular_solve(self.lane(col), &mut x);
+            x.sort_unstable();
+            for &row in &x {
+                assert!(sp.insert(col, row).is_ok());
+            }
+        }
+        sp.build()
+    }
+}
+
+/// A builder that allows for constructing a sparsity pattern.
+/// It requires elements to be added in sorted order. Specifically,
+/// For each element the major must be >= the previous element's major.
+/// If the major is the same, the minor must be in ascending order.
+#[derive(Debug, Clone, PartialEq, Eq)]
+pub struct SparsityPatternBuilder {
+    buf: SparsityPattern,
+    major_dim: usize,
+}
+
+/// An error when adding into the SparsityPatternBuilder
+#[derive(Debug, Copy, Clone, PartialEq, Eq)]
+pub enum BuilderInsertError {
+    ///
+    MajorTooLow,
+    ///
+    MinorTooLow,
+}
+
+impl SparsityPatternBuilder {
+    /// Constructs a new empty builder.
+    pub fn new(major_dim: usize, minor_dim: usize) -> Self {
+        Self {
+            buf: SparsityPattern {
+                major_offsets: vec![0],
+                minor_indices: vec![],
+                minor_dim,
+            },
+            major_dim,
+        }
+    }
+    /// The number of non-zero entries inserted into `self`.
+    pub fn num_entries(&self) -> usize {
+        self.buf.minor_indices.len()
+    }
+
+    /// Allows for general assignment of indices
+    pub fn insert(&mut self, maj: usize, min: usize) -> Result<(), BuilderInsertError> {
+        assert!(maj < self.major_dim);
+        assert!(min < self.buf.minor_dim);
+
+        let curr_major = self.buf.major_dim();
+
+        // cannot go backwards in major
+        if maj < curr_major {
+            return Err(BuilderInsertError::MajorTooLow);
+        }
+        // cannot go backwards in minor
+        if maj == curr_major
+            && *self.buf.major_offsets.last().unwrap() < self.buf.minor_indices.len()
+            && !self.buf.minor_indices.is_empty()
+            && min <= *self.buf.minor_indices.last().unwrap()
+        {
+            return Err(BuilderInsertError::MinorTooLow);
+        }
+        // add any advances in row.
+        for _ in curr_major..maj {
+            self.buf.major_offsets.push(self.buf.minor_indices.len());
+        }
+        self.buf.minor_indices.push(min);
+        Ok(())
+    }
+    /// Returns a valid partial sparsity pattern.
+    /// All the major lanes up to the current insertion will be completed.
+    pub(crate) fn valid_partial(&mut self) -> &SparsityPattern {
+        if *self.buf.major_offsets.last().unwrap() != self.buf.minor_indices.len() {
+            self.buf.major_offsets.push(self.buf.minor_indices.len());
+        }
+        &self.buf
+    }
+    /// Consumes self and outputs the constructed `SparsityPattern`.
+    /// If elements were added to the last major, but `advance_major`
+    /// was not called, will implicitly call `advance_major` then
+    /// output the values.
+    #[inline]
+    pub fn build(mut self) -> SparsityPattern {
+        self.buf
+            .major_offsets
+            .resize(self.major_dim + 1, self.buf.minor_indices.len());
+        assert_eq!(self.buf.major_dim(), self.major_dim);
+        self.buf
+    }
+    /// Returns the current major being modified by `self`.
+    pub fn current_major(&self) -> usize {
+        assert!(!self.buf.major_offsets.is_empty());
+        self.buf.major_offsets.len() - 1
+    }
+
+    /// Reverts the major index of `self` back to `maj`, deleting any entries ahead of it.
+    /// Preserves entries in `maj`.
+    pub fn revert_to_major(&mut self, maj: usize) -> bool {
+        // preserve maj + 1 elements in self
+        if self.buf.major_offsets.len() + 1 <= maj {
+            return false;
+        }
+        let last = self.buf.major_offsets[maj + 1];
+        self.buf.major_offsets.truncate(maj + 1);
+        self.buf.minor_indices.truncate(last + 1);
+        true
+    }
+
+    /// Allows for rebuilding part of a sparsity pattern, assuming that
+    /// items after maj_start have not been filled in.
+    pub fn from(sp: SparsityPattern) -> Self {
+        SparsityPatternBuilder {
+            major_dim: sp.major_dim(),
+            buf: sp,
+        }
+    }
 }
 
 impl Default for SparsityPattern {
diff --git a/nalgebra-sparse/tests/lu_factorization.rs b/nalgebra-sparse/tests/lu_factorization.rs
new file mode 100644
index 00000000..df5dd7b0
--- /dev/null
+++ b/nalgebra-sparse/tests/lu_factorization.rs
@@ -0,0 +1,46 @@
+use nalgebra_sparse::factorization::LeftLookingLUFactorization;
+use nalgebra_sparse::CscBuilder;
+
+#[test]
+fn test_basic_lu_factorization() {
+    let n = 5;
+    let mut a = CscBuilder::new(n, n);
+    for i in 0..n {
+        assert!(a.insert(i, i, 1.).is_ok());
+    }
+    // construct an identity matrix as a basic test
+    let a = a.build();
+
+    let lu_fact = LeftLookingLUFactorization::new(&a);
+
+    assert_eq!(lu_fact.u(), a);
+}
+
+#[test]
+fn test_basic_lu_factorization_with_one_more_entry() {
+    let n = 3;
+    let mut a = CscBuilder::new(n, n);
+    for i in 0..n {
+        assert!(a.insert(i, i, if i == 0 { 0.5 } else { 1. }).is_ok());
+        if i == 0 {
+            assert!(a.insert(1, 0, 1.).is_ok());
+        }
+    }
+    // construct an identity matrix as a basic test
+    let a = a.build();
+
+    let lu_fact = LeftLookingLUFactorization::new(&a);
+
+    let mut ground_truth = CscBuilder::new(n, n);
+    for i in 0..n {
+        assert!(ground_truth
+            .insert(i, i, if i == 0 { 0.5 } else { 1. })
+            .is_ok());
+        if i == 0 {
+            assert!(ground_truth.insert(1, 0, 2.).is_ok());
+        }
+    }
+    let gt = ground_truth.build();
+
+    assert_eq!(lu_fact.lu(), &gt);
+}
diff --git a/nalgebra-sparse/tests/sparsity.rs b/nalgebra-sparse/tests/sparsity.rs
new file mode 100644
index 00000000..5029e1ee
--- /dev/null
+++ b/nalgebra-sparse/tests/sparsity.rs
@@ -0,0 +1,131 @@
+use nalgebra_sparse::pattern::{SparsityPattern, SparsityPatternBuilder};
+
+#[test]
+fn sparsity_identity() {
+    let n = 100;
+    let speye = SparsityPattern::identity(n);
+    for (i, j) in speye.entries() {
+        assert_eq!(i, j);
+    }
+    assert_eq!(speye.major_dim(), n);
+    assert_eq!(speye.minor_dim(), n);
+}
+
+#[test]
+fn lower_sparse_solve() {
+    // just a smaller number so it's easier to debug
+    let n = 8;
+    let speye = SparsityPattern::identity(n);
+    let mut buf = vec![];
+    speye.sparse_lower_triangular_solve(&[0, 5], &mut buf);
+    assert_eq!(buf, vec![0, 5]);
+
+    // test case from
+    // https://www.youtube.com/watch?v=1dGRTOwBkQs&ab_channel=TimDavis
+    let mut builder = SparsityPatternBuilder::new(14, 14);
+    // building CscMatrix, so it will be col, row
+    #[rustfmt::skip]
+    let indices = vec![
+      (0, 0), (0, 2),
+      (1, 1), (1, 3), (1, 6), (1, 8),
+      (2,2), (2,4), (2,7),
+      (3,3), (3,8),
+      (4,4), (4,7),
+      (5,5), (5,8), (5,9),
+      (6,6), (6,9), (6,10),
+      (7,7), (7,9),
+      (8,8), (8,11), (8,12),
+      (9,9), (9,10), (9, 12), (9, 13),
+      (10,10), (10,11), (10,12),
+      (11,11), (11,12),
+      (12,12), (12,13),
+      (13,13),
+    ];
+    for (maj, min) in indices.iter().copied() {
+        assert!(builder.insert(maj, min).is_ok());
+    }
+    let sp = builder.build();
+    assert_eq!(sp.major_dim(), 14);
+    assert_eq!(sp.minor_dim(), 14);
+    assert_eq!(sp.nnz(), indices.len());
+    for ij in sp.entries() {
+        assert!(indices.contains(&ij));
+    }
+    sp.sparse_lower_triangular_solve(&[3, 5], &mut buf);
+    assert_eq!(buf, vec![3, 8, 11, 12, 13, 5, 9, 10]);
+}
+
+// this test is a flipped version of lower sparse solve
+#[test]
+fn upper_sparse_solve() {
+    // just a smaller number so it's easier to debug
+    let n = 8;
+    let speye = SparsityPattern::identity(n);
+    let mut buf = vec![];
+    speye.sparse_lower_triangular_solve(&[0, 5], &mut buf);
+    assert_eq!(buf, vec![0, 5]);
+
+    // test case from
+    // https://www.youtube.com/watch?v=1dGRTOwBkQs&ab_channel=TimDavis
+    let mut builder = SparsityPatternBuilder::new(14, 14);
+    // building CscMatrix, so it will be col, row
+    #[rustfmt::skip]
+    let mut indices = vec![
+      (0, 0), (0, 2),
+      (1, 1), (1, 3), (1, 6), (1, 8),
+      (2,2), (2,4), (2,7),
+      (3,3), (3,8),
+      (4,4), (4,7),
+      (5,5), (5,8), (5,9),
+      (6,6), (6,9), (6,10),
+      (7,7), (7,9),
+      (8,8), (8,11), (8,12),
+      (9,9), (9,10), (9, 12), (9, 13),
+      (10,10), (10,11), (10,12),
+      (11,11), (11,12),
+      (12,12), (12,13),
+      (13,13),
+    ];
+    indices.sort_by_key(|&(min, maj)| (maj, min));
+    for (min, maj) in indices.iter().copied() {
+        assert!(builder.insert(maj, min).is_ok());
+    }
+    let sp = builder.build();
+    assert_eq!(sp.major_dim(), 14);
+    assert_eq!(sp.minor_dim(), 14);
+    assert_eq!(sp.nnz(), indices.len());
+    sp.sparse_upper_triangular_solve(&[9], &mut buf);
+    assert_eq!(buf, vec![9, 7, 4, 2, 0, 6, 1, 5]);
+}
+
+#[test]
+fn test_builder() {
+    let mut builder = SparsityPatternBuilder::new(2, 2);
+    assert!(builder.insert(0, 0).is_ok());
+    assert!(builder.insert(0, 0).is_err());
+    assert!(builder.insert(0, 1).is_ok());
+    assert!(builder.insert(0, 1).is_err());
+    assert!(builder.insert(1, 0).is_ok());
+    assert!(builder.insert(1, 0).is_err());
+}
+
+#[test]
+fn test_builder_reset() {
+    let mut builder = SparsityPatternBuilder::new(4, 4);
+    for i in 0..3 {
+        assert!(builder.insert(i, i + 1).is_ok());
+    }
+    let out = builder.build();
+    assert_eq!(out.major_dim(), 4);
+    assert_eq!(out.minor_dim(), 4);
+    let mut builder = SparsityPatternBuilder::from(out);
+    for i in (0..=2).rev() {
+        assert!(builder.revert_to_major(i));
+        assert_eq!(builder.current_major(), i);
+    }
+    let out = builder.build();
+
+    let mut builder = SparsityPatternBuilder::from(out);
+    assert!(builder.revert_to_major(1));
+    assert_eq!(builder.current_major(), 1);
+}
diff --git a/nalgebra-sparse/tests/unit_tests/cholesky.rs b/nalgebra-sparse/tests/unit_tests/cholesky.rs
index 82cb2c42..11467c25 100644
--- a/nalgebra-sparse/tests/unit_tests/cholesky.rs
+++ b/nalgebra-sparse/tests/unit_tests/cholesky.rs
@@ -9,7 +9,7 @@ use nalgebra::proptest::matrix;
 use proptest::prelude::*;
 use matrixcompare::{assert_matrix_eq, prop_assert_matrix_eq};
 
-fn positive_definite() -> impl Strategy<Value=CscMatrix<f64>> {
+pub fn positive_definite() -> impl Strategy<Value=CscMatrix<f64>> {
     let csc_f64 = csc(value_strategy::<f64>(),
                       PROPTEST_MATRIX_DIM,
                       PROPTEST_MATRIX_DIM,
@@ -114,4 +114,4 @@ fn test_cholesky(a: Matrix5<f64>) {
     let l = DMatrix::from_iterator(l.nrows(), l.ncols(), l.iter().cloned());
     let cs_l_mat = DMatrix::from(&cs_l);
     assert_matrix_eq!(l, cs_l_mat, comp = abs, tol = 1e-12);
-}
\ No newline at end of file
+}
diff --git a/nalgebra-sparse/tests/unit_tests/csc.rs b/nalgebra-sparse/tests/unit_tests/csc.rs
index b95f048f..a54a5b74 100644
--- a/nalgebra-sparse/tests/unit_tests/csc.rs
+++ b/nalgebra-sparse/tests/unit_tests/csc.rs
@@ -733,3 +733,110 @@ proptest! {
         prop_assert_eq!(DMatrix::from(&csc), DMatrix::identity(n, n));
     }
 }
+
+#[test]
+fn test_dense_lower_triangular_solve() {
+    let mut a = CscMatrix::identity(3);
+    let v = [1., 2., 3.];
+    let mut out = [0.; 3];
+    a.dense_lower_triangular_solve(&v, &mut out, true);
+    assert_eq!(out, v);
+    a.dense_lower_triangular_solve(&v, &mut out, false);
+    assert_eq!(out, v);
+    if let SparseEntryMut::NonZero(e) = a.index_entry_mut(0, 0) {
+        *e = 2.;
+    };
+    a.dense_lower_triangular_solve(&v, &mut out, false);
+    assert_eq!(out, [0.5, 2., 3.]);
+    a.dense_lower_triangular_solve(&v, &mut out, true);
+    assert_eq!(out, v);
+}
+
+#[test]
+fn test_sparse_lower_triangular_solve() {
+    let a = CscMatrix::identity(3);
+    let vi = [0, 1, 2];
+    let v = [1., 2., 3.];
+    let mut out = [0.; 3];
+    a.sparse_lower_triangular_solve(&vi, &v, &vi, &mut out, true);
+    assert_eq!(out, v);
+    a.sparse_lower_triangular_solve(&vi, &v, &vi, &mut out, false);
+    assert_eq!(out, v);
+
+    let vi = [0, 999];
+    let v = [3., 2.];
+    let mut out = [0.; 2];
+
+    // test with large identity matrix
+    let mut a = CscMatrix::identity(1000);
+    a.sparse_lower_triangular_solve(&vi, &v, &vi, &mut out, true);
+    assert_eq!(out, v);
+    a.sparse_lower_triangular_solve(&vi, &v, &vi, &mut out, false);
+    assert_eq!(out, v);
+
+    if let SparseEntryMut::NonZero(e) = a.index_entry_mut(0, 0) {
+        *e = 2.;
+    };
+    a.sparse_lower_triangular_solve(&vi, &v, &vi, &mut out, false);
+    assert_eq!(out, [1.5, 2.]);
+    a.sparse_lower_triangular_solve(&vi, &v, &vi, &mut out, true);
+    assert_eq!(out, v);
+
+    use nalgebra_sparse::coo::CooMatrix;
+    let a: CscMatrix<f32> = (&CooMatrix::<f32>::try_from_triplets(
+        1000,
+        1000,
+        vec![0, 999, 999],
+        vec![0, 0, 999],
+        vec![2., 1., 4.],
+    )
+    .unwrap())
+        .into();
+    assert_eq!(a.col(0).nnz(), 2);
+
+    a.sparse_lower_triangular_solve(&vi, &v, &vi, &mut out, false);
+    assert_eq!(out, [1.5, 0.125]);
+    a.sparse_lower_triangular_solve(&vi, &v, &vi, &mut out, true);
+    assert_eq!(out, [3., -1.]);
+}
+
+#[test]
+fn test_sparse_upper_triangular_solve() {
+    let a = CscMatrix::identity(3);
+    let vi = [0, 1, 2];
+    let v = [1., 2., 3.];
+    let mut out = [0.; 3];
+    a.sparse_upper_triangular_solve_sorted(&vi, &v, &vi, &mut out);
+    assert_eq!(out, v);
+
+    let vi = [0, 999];
+    let v = [3., 2.];
+    let mut out = [0.; 2];
+
+    // test with large identity matrix
+    let mut a = CscMatrix::identity(1000);
+    a.sparse_upper_triangular_solve_sorted(&vi, &v, &vi, &mut out);
+    assert_eq!(out, v);
+
+    if let SparseEntryMut::NonZero(e) = a.index_entry_mut(0, 0) {
+        *e = 2.;
+    };
+    a.sparse_upper_triangular_solve_sorted(&vi, &v, &vi, &mut out);
+    assert_eq!(out, [1.5, 2.]);
+
+    use nalgebra_sparse::coo::CooMatrix;
+    let a: CscMatrix<f32> = (&CooMatrix::<f32>::try_from_triplets(
+        5,
+        5,
+        vec![0, 0, 4],
+        vec![0, 4, 4],
+        vec![2., 1., 4.],
+    )
+    .unwrap())
+        .into();
+    assert_eq!(a.col(0).nnz(), 1);
+    assert_eq!(a.col(4).nnz(), 2);
+
+    a.sparse_upper_triangular_solve_sorted(&[0, 4], &v, &[0, 4], &mut out);
+    assert_eq!(out, [1.5, 0.5]);
+}
diff --git a/nalgebra-sparse/tests/unit_tests/csr.rs b/nalgebra-sparse/tests/unit_tests/csr.rs
index b2312932..f5e457d6 100644
--- a/nalgebra-sparse/tests/unit_tests/csr.rs
+++ b/nalgebra-sparse/tests/unit_tests/csr.rs
@@ -518,6 +518,38 @@ fn csr_matrix_get_index_entry() {
     }
 }
 
+#[test]
+fn csr_upper_triangle_solve_ok() {
+    const N: usize = 10;
+    let eye = CsrMatrix::<f32>::identity(N);
+    let b = DMatrix::from_row_slice(N, 1, &[0.5; N]);
+    assert_eq!(eye.solve_upper_triangular(&b), Some(b));
+
+    let mut eye = CsrMatrix::<f32>::identity(N);
+    let v = if let SparseEntryMut::NonZero(v) = eye.index_entry_mut(0, 0) {
+        v
+    } else {
+        unreachable!();
+    };
+    *v = 2.0;
+    let b = DMatrix::from_row_slice(N, 1, &[1.0; N]);
+    let mut x = b.clone();
+    x[0] = 0.5;
+    assert_eq!(eye.solve_upper_triangular(&b), Some(x));
+
+    #[rustfmt::skip]
+    let dense = DMatrix::from_row_slice(3, 3, &[
+        1., 1., 1.,
+        0., 1., 1.,
+        0., 0., 1.,
+    ]);
+    let csr = CsrMatrix::from(&dense);
+
+    let b = DMatrix::from_row_slice(3, 1, &[1.0; 3]);
+    let x = DMatrix::from_row_slice(3, 1, &[-1., 0., 1.]);
+    assert_eq!(csr.solve_upper_triangular(&b), Some(x));
+}
+
 #[test]
 fn csr_matrix_row_iter() {
     #[rustfmt::skip]
diff --git a/nalgebra-sparse/tests/unit_tests/left_looking_lu.proptest-regressions b/nalgebra-sparse/tests/unit_tests/left_looking_lu.proptest-regressions
new file mode 100644
index 00000000..6b746b47
--- /dev/null
+++ b/nalgebra-sparse/tests/unit_tests/left_looking_lu.proptest-regressions
@@ -0,0 +1,7 @@
+# Seeds for failure cases proptest has generated in the past. It is
+# automatically read and these particular cases re-run before any
+# novel cases are generated.
+#
+# It is recommended to check this file in to source control so that
+# everyone who runs the test benefits from these saved cases.
+cc b8e49f3fa66a3568e1c29a47de5f3d2f67866d59b39329e2a0da1537a9c6d584 # shrinks to matrix = CscMatrix { cs: CsMatrix { sparsity_pattern: SparsityPattern { major_offsets: [0, 4, 8, 11, 16, 21], minor_indices: [0, 1, 3, 4, 0, 1, 3, 4, 2, 3, 4, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4], minor_dim: 5 }, values: [1.0, 0.0, 0.0, 0.0, 0.0, 8.445226477445164, 0.0, 5.072108001361104, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 5.072108001361104, 0.0, 0.0, 4.455405910808415] } }
diff --git a/nalgebra-sparse/tests/unit_tests/left_looking_lu.rs b/nalgebra-sparse/tests/unit_tests/left_looking_lu.rs
new file mode 100644
index 00000000..dfc52dcd
--- /dev/null
+++ b/nalgebra-sparse/tests/unit_tests/left_looking_lu.rs
@@ -0,0 +1,75 @@
+use crate::unit_tests::cholesky::positive_definite;
+
+use nalgebra_sparse::factorization::LeftLookingLUFactorization;
+
+use matrixcompare::{assert_matrix_eq, prop_assert_matrix_eq};
+use proptest::prelude::*;
+
+use crate::common::value_strategy;
+use nalgebra::proptest::matrix;
+use nalgebra_sparse::CscMatrix;
+
+proptest! {
+  // Note that positive definite matrices are guaranteed to be invertible.
+  // That's why they're used here, but it's not necessary for a matrix to be pd to be
+  // invertible.
+  #[test]
+  fn lu_for_positive_def_matrices(
+    matrix in positive_definite()
+  ) {
+    let lu = LeftLookingLUFactorization::new(&matrix);
+
+    let l = lu.l();
+    prop_assert!(l.triplet_iter().all(|(i, j, _)| j <= i));
+    let u = lu.u();
+    prop_assert!(u.triplet_iter().all(|(i, j, _)| j >= i));
+    prop_assert_matrix_eq!(l * u, matrix, comp = abs, tol = 1e-7);
+  }
+
+  #[test]
+  fn lu_solve_correct_for_positive_def_matrices(
+    (matrix, b) in positive_definite()
+      .prop_flat_map(|csc| {
+        let rhs = matrix(value_strategy::<f64>(), csc.nrows(), 1);
+        (Just(csc), rhs)
+      })
+  ) {
+    let lu = LeftLookingLUFactorization::new(&matrix);
+
+    let l = lu.l();
+    prop_assert!(l.triplet_iter().all(|(i, j, _)| j <= i));
+
+    let mut x_l = b.clone_owned();
+    l.dense_lower_triangular_solve(b.as_slice(), x_l.as_mut_slice(), true);
+
+    prop_assert_matrix_eq!(l * x_l, b, comp=abs, tol=1e-12);
+
+    let u = lu.u();
+    prop_assert!(u.triplet_iter().all(|(i, j, _)| j >= i));
+
+    let mut x_u = b.clone_owned();
+    u.dense_upper_triangular_solve(b.as_slice(), x_u.as_mut_slice());
+    prop_assert_matrix_eq!(u * x_u, b, comp = abs, tol = 1e-7);
+
+    let x = lu.solve(b.as_slice());
+    prop_assert_matrix_eq!(&matrix * &x, b, comp=abs, tol=1e-12);
+  }
+}
+
+#[test]
+fn minimized_lu() {
+    let major_offsets = vec![0, 1, 3, 4, 5, 8];
+    let minor_indices = vec![0, 1, 4, 2, 3, 1, 2, 4];
+    let values = vec![1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 3.0, 1.0];
+    assert_eq!(minor_indices.len(), values.len());
+    let mat = CscMatrix::try_from_unsorted_csc_data(5, 5, major_offsets, minor_indices, values);
+    let mat = mat.unwrap();
+
+    let lu = LeftLookingLUFactorization::new(&mat);
+
+    let l = lu.l();
+    assert!(l.triplet_iter().all(|(i, j, _)| j <= i));
+    let u = lu.u();
+    assert!(u.triplet_iter().all(|(i, j, _)| j >= i));
+    assert_matrix_eq!(l * u, mat, comp = abs, tol = 1e-7);
+}
diff --git a/nalgebra-sparse/tests/unit_tests/mod.rs b/nalgebra-sparse/tests/unit_tests/mod.rs
index 7090a493..9a74f87b 100644
--- a/nalgebra-sparse/tests/unit_tests/mod.rs
+++ b/nalgebra-sparse/tests/unit_tests/mod.rs
@@ -1,4 +1,7 @@
+// factorization tests
 mod cholesky;
+mod left_looking_lu;
+
 mod convert_serial;
 mod coo;
 mod csc;