Question: In what century did Olaf Geirstad-Alf  and Halfdan the Black live in?

full_context: Document 1: Sigtryg Eysteinsson: Sigtryg Eysteinsson was king of the Norwegian petty kingdoms Raumarike and Hedmark in what is today south-eastern Norway. He was killed by Halfdan the Black ("Halvdan Svarte") in the middle of the 9th century. Halfdan the Black subdued the area by first defeating and killing Sigtryg in battle and then defeated Sigtryg's brother and successor Eystein Eysteinsson in a series of battles.
Document 2: Halfdan Haraldsson the Black: Halfdan Haraldsson or Halfdan the Black (not to be confused with his grandfather and namesake) was a son of Harald I of Norway by his first wife, Åsa, the daughter of Jarl Håkon Grjotgardsson of Lade. He was made sub-king of the Trondelag by his father, along with his brother Halfdan the White. According to "Heimskringla", Halfdan the Black was poisoned, possibly at the behest of his sister in law Gunnhild, Mother of Kings.
Document 3: Olaf Guthfrithson: Olaf Guthfrithson (Old Norse: "Óláfr Guðrøðsson" ; ; Old Irish: "Amlaíb mac Gofraid" ; died 941) was a Viking leader who ruled Dublin and Viking Northumbria in the 10th century. He was the son of Gofraid ua Ímair and great-grandson of Ímar, making him one of the Uí Ímair. Olaf succeeded his father as King of Dublin in 934 and succeeded in establishing dominance over the Vikings of Limerick when he captured their king, Amlaíb Cenncairech, in 937. That same year he allied with Constantine II of Scotland in an attempt to reclaim the Kingdom of Northumbria which his father had ruled briefly in 927. The forces of Olaf and Constantine were defeated by the English led by Æthelstan at the Battle of Brunanburh.
Document 4: Harald Bluetooth: Harald "Bluetooth" Gormsson (Old Norse: "Haraldr Gormsson" , Danish: "Harald Blåtand Gormsen" , died c. 985/86) was a king of Denmark and Norway.
Document 5: Halfdan Ragnarsson: Halfdan Ragnarsson (Old Norse: "Hálfdan" ; or "Healfdene"; Old Irish: "Albann" ; died 877) was a Viking leader and a commander of the Great Heathen Army which invaded the Anglo-Saxon kingdoms of England, starting in 865. According to the tradition recorded in the Norse sagas he was one of the sons of Ragnar Lodbrok, and his brothers included Björn Ironside, Ivar the Boneless, Sigurd Snake-in-the-Eye and Ubba. He was the first Viking King of Northumbria and a pretender to the throne of Kingdom of Dublin. He died at the Battle of Strangford Lough in 877 trying to press his Irish claim.
Document 6: Ragnvald Heidumhære: Ragnvald Heidumhære (or Rognvald) was a legendary, possibly historic, petty king or chieftain of Vestfold in what is today Norway in the 9th century, according to "Ynglingatal" and to "Ynglinga saga" in "Heimskringla". He was the son of Ragnar Lodbrok or Olaf Geirstad-Alf. His name "Heiðumhæri" could be translated as "highly honoured"
Document 7: Halfdan the Black: Halfdan the Black (Old Norse: "Halfdanr Svarti") (  810 –  860 ) was a ninth-century king of Vestfold. He belonged to the House of Yngling and was the father of Harald Fairhair, the first king of Norway.
Document 8: Legendary Saga of St. Olaf: The Legendary Saga of St. Olaf or Helgisaga Óláfs konungs Haraldssonar is one of the kings' sagas, a 13th-century biography of the 11th-century Saint Olaf II of Norway. It is based heavily on the largely lost "Oldest Saga of St. Olaf". The composition is primitive and clumsy and the saga essentially consists of a series of separate anecdotes extracted from skaldic verse. The anonymous author may have been a Norwegian and the saga is preserved in one mid-13th-century Norwegian manuscript. It is thought to have been composed in the early 13th century. Snorri Sturluson is believed to have used a work closely similar to the "Legendary Saga" when he composed his "Separate Saga of St. Olaf" and the "Heimskringla".
Document 9: Þáttr Ólafs Geirstaða Alfs: "Þáttr Ólafs Geirstaða Alfs" is a "þáttr ", a short anecdotal story, about the mythical Norse king Olaf Geirstad-Alf. It is preserved in "Flateyjarbók", a saga collection that was written at the end of the 14th century. Olaf Geirstad Alf, who is also one of the kings who appear in the poem "Ynglingatal", is said to have been king of Vestfold or Grenland in Norway in the late 9th century.
Document 10: Geirmund the Noisy: Geirmund the Noisy (died c. 978) was a Viking adventurer of the 10th century. Around 975, he hosted Olaf the Peacock during the latter's second expedition to Norway. On his return home to Iceland Olaf brought Geirmund with him and Geirmund fell in love with Olaf's daughter Thurid. Though Olaf was opposed to the match, Geirmund bribed Thorgerd to be his advocate, and Olaf relented. The marriage was an unhappy one, and after three years Geirmund decided to return home without leaving any money for the support of his ex-wife and daughter. Enraged, Thurid boarded his ship before he departed, stole his famous sword "Leg-Biter," and left their infant daughter Groa on the ship. Geirmund cursed the sword, and on his return to Norway he and all of his shipmates, including little Groa, were drowned.
Document 11: Olaf Geirstad-Alf: Olaf Gudrødsson, or as he was named after his death Olaf Geirstad-Alf (Old Norse Ólaf Geirstaða Álfr), was a legendary Norwegian king of the House of Yngling from the "Ynglinga saga". He was the son of Gudrød the Hunter and according to the late "Heimskringla", the half-brother of Halfdan the Black. Gudrød and Olaf conquered a large part of Raumarike.
Document 12: Olaf the Peacock: Olaf the Peacock (Old Norse: Ólafr Pái) or Olaf Hoskuldsson (Ólafr Höskuldsson) (c. 938–1006) was a merchant and chieftain of the early Icelandic Commonwealth, who was nicknamed "the Peacock" because of his proud bearing and magnificent wardrobe. He is a major character in the "Laxdæla saga" and is mentioned in a number of other Icelandic sources. The son of a slave woman, Olaf became one of the wealthiest landowners in Iceland and played a major role in its politics and society during the latter half of the tenth century. In addition to the "Laxdæla Saga" in which he takes a leading role, Olaf also is mentioned in "Egils saga", "Njáls saga", "Gunnlaugs saga", "Kormáks saga", "Grettirs saga" and the "Landnámabók", among others.
Document 13: Gryting, King of Orkdal: Gryting was according to Heimskringla a king of Orkdal in the period leading up to Harald Fairhair's unification of Norway. He was the first leader in Trøndelag to make a stand against King Harald's ravaging of the country. At the battle of Orkdal Gryting's men were defeated and Gryting taken prisoner. After this Gryting swore allegiance to king Harald.
Document 14: Amlaíb Conung: Amlaíb Conung (Old Norse: "Óláfr" ; died c. 874) was a Viking leader in Ireland and Scotland in the mid-late ninth century. He was the son of the king of Lochlann, identified in the non-contemporary "Fragmentary Annals of Ireland" as Gofraid, and brother of Auisle and Ímar, the latter of whom founded the Uí Ímair dynasty, and whose descendants would go on to dominate the Irish Sea region for several centuries. Another Viking leader, Halfdan Ragnarsson, is considered by some scholars to be another brother. The Irish Annals title Amlaíb, Ímar and Auisle "kings of the foreigners". Modern scholars use the title "kings of Dublin" after the Viking settlement which formed the base of their power. The epithet "Conung" is derived from the Old Norse "konungr" and simply means "king". Some scholars consider Amlaíb to be identical to Olaf the White, a Viking sea-king who features in the "Landnámabók" and other Icelandic sagas.
Document 15: Olvir Hnufa: Olvir Hnufa or Ölvir hnúfa was a Norwegian hersir and skald of the late ninth and early tenth centuries, known from, among other sources, "Egil's Saga", "Skaldatal" and the "Prose Edda". Olvir was the son of the viking Berle-Kari and brother-in-law of Kveldulf Bjalfason, who married Olvir's sister Salbjorg Karadottir; he was thus uncle to Skallagrim and Thorolf Kveldulfsson and great uncle to the famous poet Egil Skallagrimsson. Olvir also had a brother named Eyvind Lambi. Olvir was a prominent member of the court of King Harald Fairhair, who united Norway under his rule in the late ninth or early tenth century. Among other famous skalds, he served as one of King Harald's court poets. He also served as a warrior in Harald's retinue, and fought at the pivotal Battle of Hafrsfjord on the king's flagship. He is best known for his involvement in the conflict between Harald and Olvir's kinsman Thorolf Kveldulfsson, which ended with the latter's death. Only a few fragments of Olvir's poetry survive.
ground_truth: ninth-century