diff --git a/cgo/lapack.go b/cgo/lapack.go
index ad34cbd3..d9c51ccd 100644
--- a/cgo/lapack.go
+++ b/cgo/lapack.go
@@ -576,6 +576,21 @@ func (impl Implementation) Dtrcon(norm lapack.MatrixNorm, uplo blas.Uplo, diag b
 	return rcond[0]
 }
 
+// Dtrtri computes the inverse of a triangular matrix, storing the result in place
+// into a. This is the BLAS level 3 version of the algorithm.
+//
+// Dtrti returns whether the matrix a is singular.
+func (impl Implementation) Dtrtri(uplo blas.Uplo, diag blas.Diag, n int, a []float64, lda int) (ok bool) {
+	checkMatrix(n, n, a, lda)
+	if uplo != blas.Upper && uplo != blas.Lower {
+		panic(badUplo)
+	}
+	if diag != blas.NonUnit && diag != blas.Unit {
+		panic(badDiag)
+	}
+	return clapack.Dtrtri(uplo, diag, n, a, lda)
+}
+
 // Dtrtrs solves a triangular system of the form A * X = B or A^T * X = B. Dtrtrs
 // returns whether the solve completed successfully. If A is singular, no solve is performed.
 func (impl Implementation) Dtrtrs(uplo blas.Uplo, trans blas.Transpose, diag blas.Diag, n, nrhs int, a []float64, lda int, b []float64, ldb int) (ok bool) {
diff --git a/cgo/lapack_test.go b/cgo/lapack_test.go
index 71676166..017e78ae 100644
--- a/cgo/lapack_test.go
+++ b/cgo/lapack_test.go
@@ -94,3 +94,7 @@ func TestDpocon(t *testing.T) {
 func TestDtrcon(t *testing.T) {
 	testlapack.DtrconTest(t, impl)
 }
+
+func TestDtrtri(t *testing.T) {
+	testlapack.DtrtriTest(t, impl)
+}
diff --git a/native/dtrti2.go b/native/dtrti2.go
new file mode 100644
index 00000000..69e1f3eb
--- /dev/null
+++ b/native/dtrti2.go
@@ -0,0 +1,51 @@
+package native
+
+import (
+	"github.com/gonum/blas"
+	"github.com/gonum/blas/blas64"
+)
+
+// Dtrti2 computes the inverse of a triangular matrix, storing the result in place
+// into a. This is the BLAS level 2 version of the algorithm.
+func (impl Implementation) Dtrti2(uplo blas.Uplo, diag blas.Diag, n int, a []float64, lda int) {
+	checkMatrix(n, n, a, lda)
+	if uplo != blas.Upper && uplo != blas.Lower {
+		panic(badUplo)
+	}
+	if diag != blas.NonUnit && diag != blas.Unit {
+		panic(badDiag)
+	}
+	bi := blas64.Implementation()
+
+	nonUnit := diag == blas.NonUnit
+	// TODO(btracey): Replace this with a row-major ordering.
+	if uplo == blas.Upper {
+		for j := 0; j < n; j++ {
+			var ajj float64
+			if nonUnit {
+				ajj = 1 / a[j*lda+j]
+				a[j*lda+j] = ajj
+				ajj *= -1
+			} else {
+				ajj = -1
+			}
+			bi.Dtrmv(blas.Upper, blas.NoTrans, diag, j, a, lda, a[j:], lda)
+			bi.Dscal(j, ajj, a[j:], lda)
+		}
+		return
+	}
+	for j := n - 1; j >= 0; j-- {
+		var ajj float64
+		if nonUnit {
+			ajj = 1 / a[j*lda+j]
+			a[j*lda+j] = ajj
+			ajj *= -1
+		} else {
+			ajj = -1
+		}
+		if j < n-1 {
+			bi.Dtrmv(blas.Lower, blas.NoTrans, diag, n-j-1, a[(j+1)*lda+j+1:], lda, a[(j+1)*lda+j:], lda)
+			bi.Dscal(n-j-1, ajj, a[(j+1)*lda+j:], lda)
+		}
+	}
+}
diff --git a/native/dtrtri.go b/native/dtrtri.go
new file mode 100644
index 00000000..73ff2da4
--- /dev/null
+++ b/native/dtrtri.go
@@ -0,0 +1,60 @@
+package native
+
+import (
+	"github.com/gonum/blas"
+	"github.com/gonum/blas/blas64"
+)
+
+// Dtrtri computes the inverse of a triangular matrix, storing the result in place
+// into a. This is the BLAS level 3 version of the algorithm which builds upon
+// Dtrti2 to operate on matrix blocks instead of only individual columns.
+//
+// Dtrti returns whether the matrix a is singular or whether it's not singular.
+// If the matrix is singular the inversion is not performed.
+func (impl Implementation) Dtrtri(uplo blas.Uplo, diag blas.Diag, n int, a []float64, lda int) (ok bool) {
+	checkMatrix(n, n, a, lda)
+	if uplo != blas.Upper && uplo != blas.Lower {
+		panic(badUplo)
+	}
+	if diag != blas.NonUnit && diag != blas.Unit {
+		panic(badDiag)
+	}
+	if n == 0 {
+		return
+	}
+	nonUnit := diag == blas.NonUnit
+	if nonUnit {
+		for i := 0; i < n; i++ {
+			if a[i*lda+i] == 0 {
+				return false
+			}
+		}
+	}
+
+	bi := blas64.Implementation()
+
+	nb := impl.Ilaenv(1, "DTRTRI", "UD", n, -1, -1, -1)
+	if nb <= 1 || nb > n {
+		impl.Dtrti2(uplo, diag, n, a, lda)
+		return true
+	}
+	if uplo == blas.Upper {
+		for j := 0; j < n; j += nb {
+			jb := min(nb, n-j)
+			bi.Dtrmm(blas.Left, blas.Upper, blas.NoTrans, diag, j, jb, 1, a, lda, a[j:], lda)
+			bi.Dtrsm(blas.Right, blas.Upper, blas.NoTrans, diag, j, jb, -1, a[j*lda+j:], lda, a[j:], lda)
+			impl.Dtrti2(blas.Upper, diag, jb, a[j*lda+j:], lda)
+		}
+		return true
+	}
+	nn := ((n - 1) / nb) * nb
+	for j := nn; j >= 0; j -= nb {
+		jb := min(nb, n-j)
+		if j+jb <= n-1 {
+			bi.Dtrmm(blas.Left, blas.Lower, blas.NoTrans, diag, n-j-jb, jb, 1, a[(j+jb)*lda+j+jb:], lda, a[(j+jb)*lda+j:], lda)
+			bi.Dtrmm(blas.Right, blas.Lower, blas.NoTrans, diag, n-j-jb, jb, -1, a[j*lda+j:], lda, a[(j+jb)*lda+j:], lda)
+		}
+		impl.Dtrti2(blas.Lower, diag, jb, a[j*lda+j:], lda)
+	}
+	return true
+}
diff --git a/native/lapack_test.go b/native/lapack_test.go
index f42aa1f1..e50f43bc 100644
--- a/native/lapack_test.go
+++ b/native/lapack_test.go
@@ -116,6 +116,14 @@ func TestDtrcon(t *testing.T) {
 	testlapack.DtrconTest(t, impl)
 }
 
+func TestDtrtri(t *testing.T) {
+	testlapack.DtrtriTest(t, impl)
+}
+
+func TestDtrti2(t *testing.T) {
+	testlapack.Dtrti2Test(t, impl)
+}
+
 func TestIladlc(t *testing.T) {
 	testlapack.IladlcTest(t, impl)
 }
diff --git a/testlapack/dtrti2.go b/testlapack/dtrti2.go
new file mode 100644
index 00000000..dada82a3
--- /dev/null
+++ b/testlapack/dtrti2.go
@@ -0,0 +1,153 @@
+package testlapack
+
+import (
+	"math"
+	"math/rand"
+	"testing"
+
+	"github.com/gonum/blas"
+	"github.com/gonum/blas/blas64"
+	"github.com/gonum/floats"
+)
+
+type Dtrti2er interface {
+	Dtrti2(uplo blas.Uplo, diag blas.Diag, n int, a []float64, lda int)
+}
+
+func Dtrti2Test(t *testing.T, impl Dtrti2er) {
+	for _, test := range []struct {
+		a    []float64
+		n    int
+		uplo blas.Uplo
+		diag blas.Diag
+		ans  []float64
+	}{
+		{
+			a: []float64{
+				2, 3, 4,
+				0, 5, 6,
+				8, 0, 8},
+			n:    3,
+			uplo: blas.Upper,
+			diag: blas.NonUnit,
+			ans: []float64{
+				0.5, -0.3, -0.025,
+				0, 0.2, -0.15,
+				8, 0, 0.125,
+			},
+		},
+		{
+			a: []float64{
+				5, 3, 4,
+				0, 7, 6,
+				10, 0, 8},
+			n:    3,
+			uplo: blas.Upper,
+			diag: blas.Unit,
+			ans: []float64{
+				5, -3, 14,
+				0, 7, -6,
+				10, 0, 8,
+			},
+		},
+		{
+			a: []float64{
+				2, 0, 0,
+				3, 5, 0,
+				4, 6, 8},
+			n:    3,
+			uplo: blas.Lower,
+			diag: blas.NonUnit,
+			ans: []float64{
+				0.5, 0, 0,
+				-0.3, 0.2, 0,
+				-0.025, -0.15, 0.125,
+			},
+		},
+		{
+			a: []float64{
+				1, 0, 0,
+				3, 1, 0,
+				4, 6, 1},
+			n:    3,
+			uplo: blas.Lower,
+			diag: blas.Unit,
+			ans: []float64{
+				1, 0, 0,
+				-3, 1, 0,
+				14, -6, 1,
+			},
+		},
+	} {
+		impl.Dtrti2(test.uplo, test.diag, test.n, test.a, test.n)
+		if !floats.EqualApprox(test.ans, test.a, 1e-14) {
+			t.Errorf("Matrix inverse mismatch. Want %v, got %v.", test.ans, test.a)
+		}
+	}
+	bi := blas64.Implementation()
+	for _, uplo := range []blas.Uplo{blas.Upper} {
+		for _, diag := range []blas.Diag{blas.NonUnit, blas.Unit} {
+			for _, test := range []struct {
+				n, lda int
+			}{
+				{3, 0},
+				{3, 5},
+			} {
+				n := test.n
+				lda := test.lda
+				if lda == 0 {
+					lda = n
+				}
+				a := make([]float64, n*lda)
+				for i := range a {
+					a[i] = rand.Float64()
+				}
+				aCopy := make([]float64, len(a))
+				copy(aCopy, a)
+				impl.Dtrti2(uplo, diag, n, a, lda)
+				if uplo == blas.Upper {
+					for i := 1; i < n; i++ {
+						for j := 0; j < i; j++ {
+							aCopy[i*lda+j] = 0
+							a[i*lda+j] = 0
+						}
+					}
+				} else {
+					for i := 1; i < n; i++ {
+						for j := i + 1; j < n; j++ {
+							aCopy[i*lda+j] = 0
+							a[i*lda+j] = 0
+						}
+					}
+				}
+				if diag == blas.Unit {
+					for i := 0; i < n; i++ {
+						a[i*lda+i] = 1
+						aCopy[i*lda+i] = 1
+					}
+				}
+				ans := make([]float64, len(a))
+				bi.Dgemm(blas.NoTrans, blas.NoTrans, n, n, n, 1, a, lda, aCopy, lda, 0, ans, lda)
+				iseye := true
+				for i := 0; i < n; i++ {
+					for j := 0; j < n; j++ {
+						if i == j {
+							if math.Abs(ans[i*lda+i]-1) > 1e-14 {
+								iseye = false
+								break
+							}
+						} else {
+							if math.Abs(ans[i*lda+j]) > 1e-14 {
+								iseye = false
+								break
+							}
+						}
+					}
+				}
+				if !iseye {
+					t.Errorf("inv(A) * A != I. Upper = %v, unit = %v, ans = %v", uplo == blas.Upper, diag == blas.Unit, ans)
+				}
+			}
+		}
+	}
+}
diff --git a/testlapack/dtrtri.go b/testlapack/dtrtri.go
new file mode 100644
index 00000000..694d494b
--- /dev/null
+++ b/testlapack/dtrtri.go
@@ -0,0 +1,89 @@
+package testlapack
+
+import (
+	"math"
+	"math/rand"
+	"testing"
+
+	"github.com/gonum/blas"
+	"github.com/gonum/blas/blas64"
+)
+
+type Dtrtrier interface {
+	Dtrconer
+	Dtrtri(uplo blas.Uplo, diag blas.Diag, n int, a []float64, lda int) bool
+}
+
+func DtrtriTest(t *testing.T, impl Dtrtrier) {
+	bi := blas64.Implementation()
+	for _, uplo := range []blas.Uplo{blas.Upper} {
+		for _, diag := range []blas.Diag{blas.NonUnit, blas.Unit} {
+			for _, test := range []struct {
+				n, lda int
+			}{
+				{3, 0},
+				{70, 0},
+				{200, 0},
+				{3, 5},
+				{70, 92},
+				{200, 205},
+			} {
+				n := test.n
+				lda := test.lda
+				if lda == 0 {
+					lda = n
+				}
+				a := make([]float64, n*lda)
+				for i := range a {
+					a[i] = rand.Float64() + 1 // This keeps the matrices well conditioned.
+				}
+				aCopy := make([]float64, len(a))
+				copy(aCopy, a)
+				impl.Dtrtri(uplo, diag, n, a, lda)
+				if uplo == blas.Upper {
+					for i := 1; i < n; i++ {
+						for j := 0; j < i; j++ {
+							aCopy[i*lda+j] = 0
+							a[i*lda+j] = 0
+						}
+					}
+				} else {
+					for i := 1; i < n; i++ {
+						for j := i + 1; j < n; j++ {
+							aCopy[i*lda+j] = 0
+							a[i*lda+j] = 0
+						}
+					}
+				}
+				if diag == blas.Unit {
+					for i := 0; i < n; i++ {
+						a[i*lda+i] = 1
+						aCopy[i*lda+i] = 1
+					}
+				}
+				ans := make([]float64, len(a))
+				bi.Dgemm(blas.NoTrans, blas.NoTrans, n, n, n, 1, a, lda, aCopy, lda, 0, ans, lda)
+				iseye := true
+				for i := 0; i < n; i++ {
+					for j := 0; j < n; j++ {
+						if i == j {
+							if math.Abs(ans[i*lda+i]-1) > 1e-4 {
+								iseye = false
+								break
+							}
+						} else {
+							if math.Abs(ans[i*lda+j]) > 1e-4 {
+								iseye = false
+								break
+							}
+						}
+					}
+				}
+				if !iseye {
+					t.Errorf("inv(A) * A != I. Upper = %v, unit = %v, n = %v, lda = %v",
+						uplo == blas.Upper, diag == blas.Unit, n, lda)
+				}
+			}
+		}
+	}
+}